# Question 8558: Point Derivative of a Degree-at-Most-Two Polynomial

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0016-08558 · q-20261007-advance-20261008-r04-derivative-0033



## Original question and givens

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


- G001 — On all R, f(t)=(-8)t²+(-5)t+(-4), allowing a2=0. Find the derivative at x=0.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-derivative](https://ocme.evemisslab.com/en/mko/mko-derivative/)





## Step-by-step derivation

S001 · impliesBefore

declared real formula on all R;derivative means the two-sided h->0 quotient limit

After

finite-step evaluation and derivative existence are separate

Conditions: none

A derivative is the two-sided limit of[f(x+h)−f(x)]/h as h→0 with h≠0.One finite evaluation is not evidence that an arbitrary function is differentiable.

Step identifiers and dependenciesRule: derivative_limit_definition_scope · Uses: G001





S002 · calculationBefore

at original x a*x := mul(-8, 0)

After

at original x a*x = 0

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

at original x a*x+b := sum(0, -5)

After

at original x a*x+b = -5

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

at original x (a*x+b)*x := mul(-5, 0)

After

at original x (a*x+b)*x = 0

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

at original x f(x) := sum(0, -4)

After

at original x f(x) = -4

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

f(t)=(-8)*t^2+(-5)*t+(-4);x=0;f(x)=-4;h!=0

After

f(x+h)-f(x)=h*(2*(-8)*(0)+(-5)+(-8)*h)

Conditions: none

Expand(x+h)²=x²+2xh+h²,cancel the original terms,and factor out h.The a2=0 lower-degree case is valid;no nonzero-leading-coefficient assumption is added.

Step identifiers and dependenciesRule: quadratic_increment_factorization · Uses: G001, S005





S007 · calculationBefore

twice quadratic coefficient := mul(2, -8)

After

twice quadratic coefficient = -16

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

twice a times x := mul(-16, 0)

After

twice a times x = 0

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

linear part of quotient := sum(0, -5)

After

linear part of quotient = -5

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

for h!=0,quotient=-5+(-8)*h

After

h->0 derivative=-5

Conditions: none

After cancelling nonzero h,the quotient error is a2*h.If a2=0 the error is zero;otherwise for any ε>0 choose δ=ε/(1+|a2|).Then0<|h|<δ makes the error<ε,establishing the two-sided limit.

Step identifiers and dependenciesRule: quadratic_two_sided_limit · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute linear part of quotient with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute twice a times x with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute at original x f(x) with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute at original x (a*x+b)*x with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute at original x a*x+b with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute at original x a*x with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

-5

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 1e3f67a129677a04e66b89a301516081039efe25bc14e40e895d84ef1ee2cc08
- candidate record SHA-256: a6c5bc09f7fd289a9fb6ee7467fd95cfcfd00844de4e4a458ae08a27f3f8c757
- semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
- question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r04-derivative-0033.json)