Question 8568: Point Derivative of a Degree-at-Most-Two Polynomial

Original question and givens

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

• G001 — On all R, f(t)=(-18)t²+(5)t+(-14), 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

Step-by-step derivation

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

finite-step evaluation and derivative existence are separate

at original x a*x := mul(-18, 0)

at original x a*x = 0

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

at original x a*x+b = 5

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

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

at original x f(x) := sum(0, -14)

at original x f(x) = -14

f(t)=(-18)*t^2+(5)*t+(-14);x=0;f(x)=-14;h!=0

f(x+h)-f(x)=h*(2*(-18)*(0)+(5)+(-18)*h)

twice quadratic coefficient := mul(2, -18)

twice quadratic coefficient = -36

twice a times x := mul(-36, 0)

twice a times x = 0

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

linear part of quotient = 5

for h!=0,quotient=5+(-18)*h

h->0 derivative=5

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.