Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.688

Derivative of \( \displaystyle - 5 \cos{\left(x - 3 \right)} \)

Problem 2.688 · easy Video

Differentiate \( \displaystyle f(x) = - 5 \cos{\left(x - 3 \right)} \).

\[ \frac{d}{d x} \left(- 5 \cos{\left(x - 3 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - 5 \frac{d}{d x} \cos{\left(x - 3 \right)} \]
constantPull out the constant factor.✓ Proved
\[ = 5 \sin{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right) \]
trigApply the derivative rule for cosine.✓ Proved
\[ = 5 \sin{\left(x - 3 \right)} \]
chain algebraApply the chain rule to the inner function. Simplify the signs.✓ Proved

Answer \( 5 \sin{\left(x - 3 \right)} \)

Top: the graph of f = -5cos(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5sin(x - 3).

✓ 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
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: pass
qwen3.6:27b-mlx: fail (style) — Step 2 uses the label 'constant' but performs the constant-multiple rule; the vocabulary requires 'constant-multiple'. Step 4 uses the label 'chain' but only differentiates the inner function (a 'derivative' step); the chain rule structure was already applied in Step 3.

Senior review claude-sonnet-5-5, 2026-10-11: pass — All steps are correct, each changes one thing, and the labels are legitimate. d/dx[-5cos(x-3)] = 5 sin(x-3), which matches the stated answer. SymPy proved every line.

qwen3.6:27b-mlx: dismiss — Pulling out a constant factor is legitimately labelled 'constant' (the vocabulary offers both 'constant' and 'constant-multiple', and either is acceptable when two labels fit), and evaluating the inner derivative x-3 -> 1 is explicitly part of a chain-rule step under the contract.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-11 — All steps are correct, each changes one thing, and the labels are legitimate. d/dx[-5cos(x-3)] = 5 sin(x-3), which matches the stated answer. SymPy proved every line.
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 2 uses the label 'constant' but performs the constant-multiple rule; the vocabulary requires 'constant-multiple'. Step 4 uses the label 'chain' but only differentiates the inner function (a 'derivative' step); the chain rule structure was already applied in Step 3.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the Derivative(x - 3, x) term). Step 4 merely evaluates the derivative of the inner function, which should be labeled 'derivative' or 'algebra', not 'chain'.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two rules at once (the derivative of cosine and the chain rule) but is labeled only as "trig". It should be split into separate steps or labeled appropriately. Step 1 also lacks a proper label; it should be "derivative".

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.