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R0016-08509 · q-20261007-advance-20261008-r04-derivative-0009
On all R, f(t)=(-4)t+(-5). Find f′(1) in this affine model.
G001 — On all R, f(t)=(-4)t+(-5). Find f′(1) in this affine model.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.
Before
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.
Rule: derivative_limit_definition_scope · Uses: G001
Before
affine m*x := mul(-4, 1)
After
affine m*x = -4
Conditions: none
Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
affine f(x) := sum(-4, -5)
After
affine f(x) = -9
Conditions: none
Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
f(t)=(-4)*t+(-5);x=1;f(x)=-9;h!=0
After
f(x+h)-f(x)=(-4)*h;quotient=-4
Conditions: none
Expand m(x+h)+b−(mx+b);the intercept and mx cancel,leaving mh.Cancel h only for h≠0,including the valid m=0 case.
Rule: affine_increment_cancellation · Uses: G001, S003
Before
for every h!=0 the quotient is -4
After
two-sided derivative at 1=-4
Conditions: none
The quotient is identically m throughout a punctured neighborhood,so its two-sided limit is m.The point value or intercept is not the derivative.
Rule: constant_quotient_limit · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute affine f(x) with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute affine m*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.-4
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c83560fa06af86d89832c2e04ca47ffa93cdcdbc15d1b190a432319de5f52cebcandidate record SHA-256: 4bcbc324557ba446f787cd4a8b2797d7112cd9fbd5c11b22f6d6f971de8cc85dsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED