Explore related knowledge and questions →
← Question derivation index · Original question batch
R0016-08579 · q-20261007-advance-20261008-r04-derivative-0079
On all R, f(t)=|t−(-9)|. Find its derivative at x=-12, or answer does not exist at the corner.
G001 — On all R, f(t)=|t−(-9)|. Find its derivative at x=-12, 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(-12, -9)
After
relative position x-c = -3
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(3, 2)
After
sign-preserving neighborhood radius = 3/2
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=-3;0<|h|<3/2
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=3/2
After
two-sided derivative at -12=-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: 8ccb41eef832672eecf480e791cb139a229071b22b4ffe4fd2631534fad6fc93candidate record SHA-256: 28702f26ea4abcd5d99ccae2877fa5a1ed865f09754c05cb2e3825cf79377b2fsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED