OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 8576: Derivative and Corner of a Shifted Absolute Value

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 43

← Question derivation index · Original question batch

R0016-08576 · q-20261007-advance-20261008-r04-derivative-0076

Original question and givens

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

  • G001 — On all R, f(t)=|t−(-12)|. 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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: derivative_limit_definition_scope · Uses: G001

S002 · calculation

Before

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

After

relative position x-c = 0

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

x=c=-12;f(x)=0;h!=0

After

quotient=|h|/h;-1 for h<0;+1 for h>0

Conditions: none

At the corner f(c+h)−f(c)=|h|.The negative-h quotient is−1 and the positive-h quotient is+1.The unequal side limits preclude a derivative;a symmetric quotient0 does not replace it.

Step identifiers and dependencies

Rule: absolute_corner_one_sided_quotients · Uses: G001, S002

S004 · implies

Before

left derivative=-1;right derivative=1

After

two-sided derivative=does not exist

Conditions: none

The corner derivative does not exist.Do not substitute a totalized API default0 for an existing derivative.

Step identifiers and dependencies

Rule: absolute_corner_nonexistence · Uses: G001, S003

Reverse checks

  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

does not exist

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: 4290d569390bff67b6912fb95000f2ed38b24d24ee5e47b6645801a422acd0cb
  • candidate record SHA-256: 078ccfc5b1bbd33b06c3892c48d6ac1e748719899622d5f7a3f22401c5abe8ab
  • semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
  • question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON