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.
On all R, f(t)=(-12)t+(3). Find f′(-2) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-11)t+(2). Find f′(-1) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-10)t+(1). Find f′(0) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-9)t+(0). Find f′(1) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-8)t+(-1). Find f′(2) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-7)t+(-2). Find f′(-2) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-6)t+(-3). Find f′(-1) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-5)t+(-4). Find f′(0) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-4)t+(-5). Find f′(1) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-3)t+(-6). Find f′(2) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-2)t+(-7). Find f′(-2) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(-1)t+(-8). Find f′(-1) in this affine model.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(0)t+(-9). Find f′(0) in this affine model.
0
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(1)t+(-10). Find f′(1) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(2)t+(-11). Find f′(2) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(3)t+(-12). Find f′(-2) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(4)t+(-13). Find f′(-1) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(5)t+(-14). Find f′(0) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(6)t+(-15). Find f′(1) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(7)t+(-16). Find f′(2) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(8)t+(-17). Find f′(-2) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(9)t+(-18). Find f′(-1) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(10)t+(-19). Find f′(0) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(11)t+(-20). Find f′(1) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(12)t+(-21). Find f′(2) in this affine model.
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
Real derivatives at a point and finite difference quotients · qf-derivative-affine-slope · candidate
On all R, f(t)=(0)t²+(-13)t+(5), allowing a2=0. Find the derivative at x=4.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-2)t²+(-11)t+(2), allowing a2=0. Find the derivative at x=-1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(3)t²+(-10)t+(1), allowing a2=0. Find the derivative at x=0.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-4)t²+(-9)t+(0), allowing a2=0. Find the derivative at x=1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(5)t²+(-8)t+(-1), allowing a2=0. Find the derivative at x=2.
12
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-6)t²+(-7)t+(-2), allowing a2=0. Find the derivative at x=-2.
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(7)t²+(-6)t+(-3), allowing a2=0. Find the derivative at x=-1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-8)t²+(-5)t+(-4), allowing a2=0. Find the derivative at x=0.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(9)t²+(-4)t+(-5), allowing a2=0. Find the derivative at x=1.
14
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-10)t²+(-3)t+(-6), allowing a2=0. Find the derivative at x=2.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(11)t²+(-2)t+(-7), allowing a2=0. Find the derivative at x=-2.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-12)t²+(-1)t+(-8), allowing a2=0. Find the derivative at x=-1.
23
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(13)t²+(0)t+(-9), allowing a2=0. Find the derivative at x=0.
0
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-14)t²+(1)t+(-10), allowing a2=0. Find the derivative at x=1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(15)t²+(2)t+(-11), allowing a2=0. Find the derivative at x=2.
62
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-16)t²+(3)t+(-12), allowing a2=0. Find the derivative at x=-2.
67
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(17)t²+(4)t+(-13), allowing a2=0. Find the derivative at x=-1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-18)t²+(5)t+(-14), allowing a2=0. Find the derivative at x=0.
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(19)t²+(6)t+(-15), allowing a2=0. Find the derivative at x=1.
44
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-20)t²+(7)t+(-16), allowing a2=0. Find the derivative at x=2.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(21)t²+(8)t+(-17), allowing a2=0. Find the derivative at x=-2.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-22)t²+(9)t+(-18), allowing a2=0. Find the derivative at x=-1.
53
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(23)t²+(10)t+(-19), allowing a2=0. Find the derivative at x=0.
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(-24)t²+(11)t+(-20), allowing a2=0. Find the derivative at x=1.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
On all R, f(t)=(25)t²+(12)t+(-21), allowing a2=0. Find the derivative at x=2.
112
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
Real derivatives at a point and finite difference quotients · qf-derivative-quadratic-point · candidate
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.
2/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
16/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
28/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
9/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
36/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
54/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
1/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
100/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
78/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
56/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
60/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
72/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
54/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
34/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
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.
132/1
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
Real derivatives at a point and finite difference quotients · qf-derivative-finite-quotient · candidate
On all R, f(t)=|t−(-12)|. Find its derivative at x=-12, or answer does not exist at the corner.
does 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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-11)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-10)|. Find its derivative at x=-8, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-9)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-8)|. Find its derivative at x=-4, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-7)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-6)|. Find its derivative at x=0, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-5)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-4)|. Find its derivative at x=4, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-3)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-2)|. Find its derivative at x=8, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(-1)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(0)|. Find its derivative at x=12, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(1)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(2)|. Find its derivative at x=16, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(3)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(4)|. Find its derivative at x=20, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(5)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(6)|. Find its derivative at x=24, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(7)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(8)|. Find its derivative at x=28, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(9)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(10)|. Find its derivative at x=32, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(11)|. Find its derivative at x=-12, or answer does not exist at the corner.
-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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
On all R, f(t)=|t−(12)|. Find its derivative at x=36, or answer does not exist at the corner.
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
Real derivatives at a point and finite difference quotients · qf-derivative-shifted-absolute · candidate
Canonical question JSONL · English JSONL · Validation report