# Real Derivative and Finite Quotient Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261007-advance-20261008-r04-derivative-foundations · 2026-10-07 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Derivative of an Affine Model

On all R, f(t)=(-12)t+(3). Find f′(-2) in this affine model.

View answer and explanation-12

The nonzero-step quotient is the slope m=-12, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0001/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 2. Derivative of an Affine Model

On all R, f(t)=(-11)t+(2). Find f′(-1) in this affine model.

View answer and explanation-11

The nonzero-step quotient is the slope m=-11, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0002/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 3. Derivative of an Affine Model

On all R, f(t)=(-10)t+(1). Find f′(0) in this affine model.

View answer and explanation-10

The nonzero-step quotient is the slope m=-10, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0003/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 4. Derivative of an Affine Model

On all R, f(t)=(-9)t+(0). Find f′(1) in this affine model.

View answer and explanation-9

The nonzero-step quotient is the slope m=-9, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0004/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 5. Derivative of an Affine Model

On all R, f(t)=(-8)t+(-1). Find f′(2) in this affine model.

View answer and explanation-8

The nonzero-step quotient is the slope m=-8, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0005/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 6. Derivative of an Affine Model

On all R, f(t)=(-7)t+(-2). Find f′(-2) in this affine model.

View answer and explanation-7

The nonzero-step quotient is the slope m=-7, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0006/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 7. Derivative of an Affine Model

On all R, f(t)=(-6)t+(-3). Find f′(-1) in this affine model.

View answer and explanation-6

The nonzero-step quotient is the slope m=-6, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0007/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 8. Derivative of an Affine Model

On all R, f(t)=(-5)t+(-4). Find f′(0) in this affine model.

View answer and explanation-5

The nonzero-step quotient is the slope m=-5, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0008/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 9. Derivative of an Affine Model

On all R, f(t)=(-4)t+(-5). Find f′(1) in this affine model.

View answer and explanation-4

The nonzero-step quotient is the slope m=-4, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0009/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 10. Derivative of an Affine Model

On all R, f(t)=(-3)t+(-6). Find f′(2) in this affine model.

View answer and explanation-3

The nonzero-step quotient is the slope m=-3, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0010/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 11. Derivative of an Affine Model

On all R, f(t)=(-2)t+(-7). Find f′(-2) in this affine model.

View answer and explanation-2

The nonzero-step quotient is the slope m=-2, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0011/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 12. Derivative of an Affine Model

On all R, f(t)=(-1)t+(-8). Find f′(-1) in this affine model.

View answer and explanation-1

The nonzero-step quotient is the slope m=-1, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0012/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 13. Derivative of an Affine Model

On all R, f(t)=(0)t+(-9). Find f′(0) in this affine model.

View answer and explanation0

The nonzero-step quotient is the slope m=0, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0013/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 14. Derivative of an Affine Model

On all R, f(t)=(1)t+(-10). Find f′(1) in this affine model.

View answer and explanation1

The nonzero-step quotient is the slope m=1, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0014/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 15. Derivative of an Affine Model

On all R, f(t)=(2)t+(-11). Find f′(2) in this affine model.

View answer and explanation2

The nonzero-step quotient is the slope m=2, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0015/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 16. Derivative of an Affine Model

On all R, f(t)=(3)t+(-12). Find f′(-2) in this affine model.

View answer and explanation3

The nonzero-step quotient is the slope m=3, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0016/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 17. Derivative of an Affine Model

On all R, f(t)=(4)t+(-13). Find f′(-1) in this affine model.

View answer and explanation4

The nonzero-step quotient is the slope m=4, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0017/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 18. Derivative of an Affine Model

On all R, f(t)=(5)t+(-14). Find f′(0) in this affine model.

View answer and explanation5

The nonzero-step quotient is the slope m=5, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0018/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 19. Derivative of an Affine Model

On all R, f(t)=(6)t+(-15). Find f′(1) in this affine model.

View answer and explanation6

The nonzero-step quotient is the slope m=6, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0019/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 20. Derivative of an Affine Model

On all R, f(t)=(7)t+(-16). Find f′(2) in this affine model.

View answer and explanation7

The nonzero-step quotient is the slope m=7, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0020/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 21. Derivative of an Affine Model

On all R, f(t)=(8)t+(-17). Find f′(-2) in this affine model.

View answer and explanation8

The nonzero-step quotient is the slope m=8, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0021/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 22. Derivative of an Affine Model

On all R, f(t)=(9)t+(-18). Find f′(-1) in this affine model.

View answer and explanation9

The nonzero-step quotient is the slope m=9, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0022/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 23. Derivative of an Affine Model

On all R, f(t)=(10)t+(-19). Find f′(0) in this affine model.

View answer and explanation10

The nonzero-step quotient is the slope m=10, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0023/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 24. Derivative of an Affine Model

On all R, f(t)=(11)t+(-20). Find f′(1) in this affine model.

View answer and explanation11

The nonzero-step quotient is the slope m=11, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0024/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 25. Derivative of an Affine Model

On all R, f(t)=(12)t+(-21). Find f′(2) in this affine model.

View answer and explanation12

The nonzero-step quotient is the slope m=12, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0025/)

Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate





## 26. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(0)t²+(-13)t+(5), allowing a2=0. Find the derivative at x=4.

View answer and explanation-13

The derivative is 2a2*x+a1=-13. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0026/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 27. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-2)t²+(-11)t+(2), allowing a2=0. Find the derivative at x=-1.

View answer and explanation-7

The derivative is 2a2*x+a1=-7. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0027/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 28. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(3)t²+(-10)t+(1), allowing a2=0. Find the derivative at x=0.

View answer and explanation-10

The derivative is 2a2*x+a1=-10. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0028/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 29. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-4)t²+(-9)t+(0), allowing a2=0. Find the derivative at x=1.

View answer and explanation-17

The derivative is 2a2*x+a1=-17. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0029/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 30. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(5)t²+(-8)t+(-1), allowing a2=0. Find the derivative at x=2.

View answer and explanation12

The derivative is 2a2*x+a1=12. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0030/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 31. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-6)t²+(-7)t+(-2), allowing a2=0. Find the derivative at x=-2.

View answer and explanation17

The derivative is 2a2*x+a1=17. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0031/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 32. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(7)t²+(-6)t+(-3), allowing a2=0. Find the derivative at x=-1.

View answer and explanation-20

The derivative is 2a2*x+a1=-20. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0032/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 33. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-8)t²+(-5)t+(-4), allowing a2=0. Find the derivative at x=0.

View answer and explanation-5

The derivative is 2a2*x+a1=-5. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0033/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 34. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(9)t²+(-4)t+(-5), allowing a2=0. Find the derivative at x=1.

View answer and explanation14

The derivative is 2a2*x+a1=14. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0034/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 35. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-10)t²+(-3)t+(-6), allowing a2=0. Find the derivative at x=2.

View answer and explanation-43

The derivative is 2a2*x+a1=-43. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0035/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 36. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(11)t²+(-2)t+(-7), allowing a2=0. Find the derivative at x=-2.

View answer and explanation-46

The derivative is 2a2*x+a1=-46. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0036/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 37. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-12)t²+(-1)t+(-8), allowing a2=0. Find the derivative at x=-1.

View answer and explanation23

The derivative is 2a2*x+a1=23. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0037/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 38. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(13)t²+(0)t+(-9), allowing a2=0. Find the derivative at x=0.

View answer and explanation0

The derivative is 2a2*x+a1=0. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0038/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 39. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-14)t²+(1)t+(-10), allowing a2=0. Find the derivative at x=1.

View answer and explanation-27

The derivative is 2a2*x+a1=-27. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0039/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 40. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(15)t²+(2)t+(-11), allowing a2=0. Find the derivative at x=2.

View answer and explanation62

The derivative is 2a2*x+a1=62. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0040/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 41. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-16)t²+(3)t+(-12), allowing a2=0. Find the derivative at x=-2.

View answer and explanation67

The derivative is 2a2*x+a1=67. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0041/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 42. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(17)t²+(4)t+(-13), allowing a2=0. Find the derivative at x=-1.

View answer and explanation-30

The derivative is 2a2*x+a1=-30. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0042/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 43. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-18)t²+(5)t+(-14), allowing a2=0. Find the derivative at x=0.

View answer and explanation5

The derivative is 2a2*x+a1=5. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0043/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 44. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(19)t²+(6)t+(-15), allowing a2=0. Find the derivative at x=1.

View answer and explanation44

The derivative is 2a2*x+a1=44. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0044/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 45. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-20)t²+(7)t+(-16), allowing a2=0. Find the derivative at x=2.

View answer and explanation-73

The derivative is 2a2*x+a1=-73. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0045/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 46. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(21)t²+(8)t+(-17), allowing a2=0. Find the derivative at x=-2.

View answer and explanation-76

The derivative is 2a2*x+a1=-76. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0046/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 47. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-22)t²+(9)t+(-18), allowing a2=0. Find the derivative at x=-1.

View answer and explanation53

The derivative is 2a2*x+a1=53. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0047/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 48. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(23)t²+(10)t+(-19), allowing a2=0. Find the derivative at x=0.

View answer and explanation10

The derivative is 2a2*x+a1=10. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0048/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 49. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(-24)t²+(11)t+(-20), allowing a2=0. Find the derivative at x=1.

View answer and explanation-37

The derivative is 2a2*x+a1=-37. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0049/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 50. Point Derivative of a Degree-at-Most-Two Polynomial

On all R, f(t)=(25)t²+(12)t+(-21), allowing a2=0. Find the derivative at x=2.

View answer and explanation112

The derivative is 2a2*x+a1=112. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0050/)

Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate





## 51. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(0)t²+(2)t+(3), allowing a2=0. At x=4 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation2/1

The specified finite quotient is 2/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0051/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 52. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-1)t²+(-11)t+(2), allowing a2=0. At x=-1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-7/1

The specified finite quotient is -7/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0052/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 53. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(2)t²+(-10)t+(4), allowing a2=0. At x=0 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-4/1

The specified finite quotient is -4/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0053/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 54. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-3)t²+(-9)t+(6), allowing a2=0. At x=1 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-12/1

The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0054/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 55. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(4)t²+(-8)t+(8), allowing a2=0. At x=2 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation16/1

The specified finite quotient is 16/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0055/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 56. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-5)t²+(-7)t+(10), allowing a2=0. At x=-2 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation28/1

The specified finite quotient is 28/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0056/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 57. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(6)t²+(-6)t+(12), allowing a2=0. At x=-1 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-12/1

The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0057/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 58. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-7)t²+(-5)t+(14), allowing a2=0. At x=0 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation9/1

The specified finite quotient is 9/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0058/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 59. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(8)t²+(-4)t+(16), allowing a2=0. At x=1 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation36/1

The specified finite quotient is 36/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0059/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 60. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-9)t²+(-3)t+(18), allowing a2=0. At x=2 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-30/1

The specified finite quotient is -30/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0060/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 61. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(10)t²+(-2)t+(20), allowing a2=0. At x=-2 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-22/1

The specified finite quotient is -22/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0061/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 62. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-11)t²+(-1)t+(22), allowing a2=0. At x=-1 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation54/1

The specified finite quotient is 54/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0062/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 63. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(12)t²+(0)t+(24), allowing a2=0. At x=0 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation12/1

The specified finite quotient is 12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0063/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 64. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-13)t²+(1)t+(26), allowing a2=0. At x=1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

The specified finite quotient is 1/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0064/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 65. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(14)t²+(2)t+(28), allowing a2=0. At x=2 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation100/1

The specified finite quotient is 100/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0065/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 66. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-15)t²+(3)t+(30), allowing a2=0. At x=-2 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation78/1

The specified finite quotient is 78/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0066/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 67. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(16)t²+(4)t+(32), allowing a2=0. At x=-1 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation4/1

The specified finite quotient is 4/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0067/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 68. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-17)t²+(5)t+(34), allowing a2=0. At x=0 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation56/1

The specified finite quotient is 56/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0068/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 69. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(18)t²+(6)t+(36), allowing a2=0. At x=1 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation60/1

The specified finite quotient is 60/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0069/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 70. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-19)t²+(7)t+(38), allowing a2=0. At x=2 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-31/1

The specified finite quotient is -31/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0070/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 71. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(20)t²+(8)t+(40), allowing a2=0. At x=-2 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-12/1

The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0071/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 72. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-21)t²+(9)t+(42), allowing a2=0. At x=-1 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation72/1

The specified finite quotient is 72/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0072/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 73. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(22)t²+(10)t+(44), allowing a2=0. At x=0 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation54/1

The specified finite quotient is 54/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0073/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 74. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(-23)t²+(11)t+(46), allowing a2=0. At x=1 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation34/1

The specified finite quotient is 34/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0074/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 75. Finite Difference Quotient at a Specified Nonzero Step

On all R, f(t)=(24)t²+(12)t+(48), allowing a2=0. At x=2 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation132/1

The specified finite quotient is 132/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0075/)

Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate





## 76. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-12)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanationdoes not exist

At the center, the left derivative is −1 and the right derivative is 1, so the derivative does not exist. Do not substitute a totalized API default of zero for existence.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0076/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 77. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-11)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0077/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 78. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-10)|. Find its derivative at x=-8, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0078/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 79. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-9)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0079/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 80. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-8)|. Find its derivative at x=-4, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0080/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 81. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-7)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0081/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 82. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-6)|. Find its derivative at x=0, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0082/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 83. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-5)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0083/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 84. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-4)|. Find its derivative at x=4, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0084/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 85. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-3)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0085/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 86. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-2)|. Find its derivative at x=8, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0086/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 87. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(-1)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0087/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 88. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(0)|. Find its derivative at x=12, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0088/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 89. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(1)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0089/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 90. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(2)|. Find its derivative at x=16, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0090/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 91. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(3)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0091/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 92. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(4)|. Find its derivative at x=20, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0092/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 93. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(5)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0093/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 94. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(6)|. Find its derivative at x=24, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0094/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 95. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(7)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0095/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 96. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(8)|. Find its derivative at x=28, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0096/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 97. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(9)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0097/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 98. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(10)|. Find its derivative at x=32, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0098/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 99. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(11)|. Find its derivative at x=-12, or answer does not exist at the corner.

View answer and explanation-1

Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0099/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate





## 100. Derivative and Corner of a Shifted Absolute Value

On all R, f(t)=|t−(12)|. Find its derivative at x=36, or answer does not exist at the corner.

View answer and explanation1

Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r04-derivative-0100/)

Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r04-derivative-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261008-r04-derivative-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r04-derivative-foundations/validation-report.json)