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R0016-08562 · q-20261007-advance-20261008-r04-derivative-0037
On all R, f(t)=(-12)t²+(-1)t+(-8), allowing a2=0. Find the derivative at x=-1.
G001 — On all R, f(t)=(-12)t²+(-1)t+(-8), allowing a2=0. Find the derivative at x=-1.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
at original x a*x := mul(-12, -1)
After
at original x a*x = 12
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
at original x a*x+b := sum(12, -1)
After
at original x a*x+b = 11
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
at original x (a*x+b)*x := mul(11, -1)
After
at original x (a*x+b)*x = -11
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, S003
Before
at original x f(x) := sum(-11, -8)
After
at original x f(x) = -19
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, S004
Before
f(t)=(-12)*t^2+(-1)*t+(-8);x=-1;f(x)=-19;h!=0
After
f(x+h)-f(x)=h*(2*(-12)*(-1)+(-1)+(-12)*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.
Rule: quadratic_increment_factorization · Uses: G001, S005
Before
twice quadratic coefficient := mul(2, -12)
After
twice quadratic coefficient = -24
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, S006
Before
twice a times x := mul(-24, -1)
After
twice a times x = 24
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, S007
Before
linear part of quotient := sum(24, -1)
After
linear part of quotient = 23
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, S008
Before
for h!=0,quotient=23+(-12)*h
After
h->0 derivative=23
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.
Rule: quadratic_two_sided_limit · Uses: G001, S009
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.23
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: ca860934dc14cab4fabbbdbf6bfc67289681887e0b74f91d5ee2909e09213e39candidate record SHA-256: b1e69ccb536fc42fd55670d034e417fdeed24bbff46a06e616706ad7143ef4dfsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED