Home›Calculus 1›Chain rule›Problem 2.106

Derivative of \( \displaystyle - \cos{\left(x - 1 \right)} \)

Problem 2.106 · medium Video

Differentiate \( \displaystyle f(x) = - \cos{\left(x - 1 \right)} \).

\[ \frac{d}{d x} \left(- \cos{\left(x - 1 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} \cos{\left(x - 1 \right)} \]
constantPull the constant factor -1 out of the derivative.✓ Proved
\[ = \sin{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) \]
trig algebraApply the derivative rule for cosine. Simplify the signs.✓ Proved
\[ = \sin{\left(x - 1 \right)} \]
derivative algebraDifferentiate the inner function x - 1. Simplify the final expression.✓ Proved

Answer \( \sin{\left(x - 1 \right)} \)

Top: the graph of f = -cos(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sin(x - 1).

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: fail (style) — Step 1 uses an unlabelled rule; it should be identified as a derivative step. All other steps are correctly labeled and apply only one rule each.
deepseek-r1:70b: fail (style) — Step 1 is missing a rule label, which is required by the contract.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-04: pass — Every step is mathematically correct and SymPy proved each one. The labels fit: constant for pulling out -1, trig for the cosine derivative, algebra for the sign cleanup, derivative for d/dx(x-1)=1, and algebra for multiplying by 1. The final line matches the stated answer sin(x - 1).

gpt-oss:20b: dismiss — Line 1 only restates the problem as a Derivative expression, which SymPy proved matches the task, so it applies no rule and needs no label.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-04 — Every step is mathematically correct and SymPy proved each one. The labels fit: constant for pulling out -1, trig for the cosine derivative, algebra for the sign cleanup, derivative for d/dx(x-1)=1, and algebra for multiplying by 1. The final line matches the stated answer sin(x - 1).
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (style) 2026-09-20 — Step 1 uses an unlabelled rule; it should be identified as a derivative step. All other steps are correctly labeled and apply only one rule each.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies two rules at once: it uses the derivative of cosine (trig) and the chain rule for the inner function x‑1. It should be split into separate steps, each labeled with a single rule from the allowed vocabulary.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — The first step lacks a rule label, which is required by the contract.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: fail (error) 2026-09-18 — Step 1 is missing the rule label, which is a defect as per the contract.
gpt-oss:20b: fail (error) 2026-09-18 — Step 3 applies the chain rule (derivative of cos(u) = -sin(u)·u') but is labeled as "trig" and combines two rules in one step. This violates the rule that each step must change only one thing and use an appropriate label.
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.