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R0016-08590 · q-20261007-advance-20261008-r04-derivative-0090
On all R, f(t)=|t−(2)|. Find its derivative at x=16, or answer does not exist at the corner.
G001 — On all R, f(t)=|t−(2)|. Find its derivative at x=16, or answer does not exist at the corner.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
relative position x-c := sub(16, 2)
After
relative position x-c = 14
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
sign-preserving neighborhood radius := div(14, 2)
After
sign-preserving neighborhood radius = 7
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
x-c=14;0<|h|<7
After
x+h-c keeps sign;quotient=1
Conditions: none
Choose half the distance to the corner as the radius.Then x+h−c has the same sign as x−c,and the absolute value uses one affine branch throughout that neighborhood,with constant corresponding quotient.
Rule: absolute_off_corner_local_sign · Uses: G001, S003
Before
local constant quotient=1;radius=7
After
two-sided derivative at 16=1
Conditions: none
The locally constant quotient gives the two-sided limit.This conclusion does not apply at the excluded corner.
Rule: absolute_off_corner_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 sign-preserving neighborhood radius with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute relative position x-c 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.1
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: ad9eb6314b9aa285b5b825822247c1014b1afe14eb22eb05f20a6e49fef76c4ecandidate record SHA-256: 600705d9f7fc90bf525c180ca8cef95c100099d36f5b9160a80c1af3ab7b7e7dsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED