# Question 8529: Finite Difference Quotient at a Specified Nonzero Step [Return to catalogue page 43](https://ocme.evemisslab.com/en/questions/page/43/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r04-derivative-foundations/#q-20261007-advance-20261008-r04-derivative-0054) 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. R0016-08529 · q-20261007-advance-20261008-r04-derivative-0054 ## Original question and givens 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. - G001 — 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. ## 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(-3, 1) After at original x a*x = -3 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(-3, -9) After at original x a*x+b = -12 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(-12, 1) After at original x (a*x+b)*x = -12 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(-12, 6) After at original x f(x) = -6 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)=(-3)*t^2+(-9)*t+(6);x=1;f(x)=-6;h!=0 After f(x+h)-f(x)=h*(2*(-3)*(1)+(-9)+(-3)*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, -3) After twice quadratic coefficient = -6 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(-6, 1) After twice a times x = -6 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(-6, -9) After linear part of quotient = -15 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 · calculationBefore shifted input x+h := sum(1, -1) After shifted input x+h = 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, S009 S011 · calculationBefore at shifted x a*x := mul(-3, 0) After at shifted 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, S010 S012 · calculationBefore at shifted x a*x+b := sum(0, -9) After at shifted x a*x+b = -9 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, S011 S013 · calculationBefore at shifted x (a*x+b)*x := mul(-9, 0) After at shifted 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, S012 S014 · calculationBefore at shifted x f(x) := sum(0, 6) After at shifted x f(x) = 6 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, S013 S015 · calculationBefore finite increment f(x+h)-f(x) := sub(6, -6) After finite increment f(x+h)-f(x) = 12 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, S014 S016 · calculationBefore specified finite quotient := div(12, -1) After specified finite quotient = -12 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, S015 S017 · calculationBefore finite-step quadratic correction := mul(-3, -1) After finite-step quadratic correction = 3 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, S016 S018 · calculationBefore expanded finite quotient := sum(-15, 3) After expanded finite quotient = -12 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, S017 S019 · calculationBefore finite quotient minus derivative := sub(-12, -15) After finite quotient minus derivative = 3 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, S018 S020 · impliesBefore h=-1;finite quotient=-12/1;derivative value=-15 After difference=3;a2*h=3 Conditions: none Keep the specified nonzero h:the answer is a finite quotient and differs from the derivative by a2*h.Special cases such as an affine model can agree numerically,but this does not identify a general finite quotient with a derivative or prove differentiability. Step identifiers and dependenciesRule: finite_difference_not_derivative_limit · Uses: G001, S019 ## Reverse checks - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute finite quotient minus derivative with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute expanded finite quotient with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute finite-step quadratic correction with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute specified finite quotient with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute finite increment f(x+h)-f(x) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute at shifted x f(x) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute at shifted x (a*x+b)*x with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute at shifted x a*x+b with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute at shifted x a*x with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute shifted input x+h with exact rational arithmetic, not rounded decimals. - 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 -12/1 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: 1152d019ecbc808e4d496f9cc4f5d118520b942b29bd363dc302220046eade32 - candidate record SHA-256: 67d18c92eb65ffa5435870a9fac62bd700efc5f0d6c6773179e97227b3d5baea - 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-0054.json)