# Question 8589: Derivative and Corner of a Shifted Absolute Value

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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.

R0016-08589 · q-20261007-advance-20261008-r04-derivative-0089



## Original question and givens

On all R, f(t)=|t−(1)|. Find its derivative at x=-12, or answer does not exist at the corner.


- G001 — On all R, f(t)=|t−(1)|. Find its derivative at x=-12, or answer does not exist at the corner.





## 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

relative position x-c := sub(-12, 1)

After

relative position x-c = -13

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

sign-preserving neighborhood radius := div(13, 2)

After

sign-preserving neighborhood radius = 13/2

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 · impliesBefore

x-c=-13;0<|h|<13/2

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.

Step identifiers and dependenciesRule: absolute_off_corner_local_sign · Uses: G001, S003





S005 · impliesBefore

local constant quotient=-1;radius=13/2

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.

Step identifiers and dependenciesRule: absolute_off_corner_limit · Uses: G001, S004









## Reverse checks


- 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.





## Result

-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: 46ec4382564156443fbccd815d488244df08a5569e82d037ac96e53570757554
- candidate record SHA-256: e0c274db51f246708f4b37af55a52132027d3627e58bdadb7e7e59eba350fb86
- 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-0089.json)