Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.833

Derivative of \( \displaystyle - \frac{\cos{\left(4 x + 1 \right)}}{2} \)

Problem 2.833 · medium Video

Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x + 1 \right)}}{2} \).

\[ \frac{d}{d x} \left(- \frac{\cos{\left(4 x + 1 \right)}}{2}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \cos{\left(4 x + 1 \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{\sin{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{2} \]
trigApply the derivative rule for cosine.✓ Proved
\[ = 2 \sin{\left(4 x + 1 \right)} \]
chain algebraApply the chain rule to the inner function. Simplify the coefficients.✓ Proved

Answer \( 2 \sin{\left(4 x + 1 \right)} \)

Top: the graph of f = -cos(4x + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2sin(4x + 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
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 3 is labeled 'trig' but applies the chain rule (Derivative(cos(u)) = -sin(u)*u'), which is not a trigonometric identity. Step 4 is labeled 'chain' but performs algebraic simplification of the inner derivative, not the chain rule itself. The labels are swapped and incorrect.

Senior review claude-sonnet-5-5, 2026-10-13: pass — The math is correct and every line was proved by SymPy. The rule labels (constant-multiple, trig, chain, algebra) each name a rule the step legitimately applied, and the final answer 2*sin(4*x+1) matches.

qwen3.6:27b-mlx: dismiss — Step 3 legitimately applies the cosine derivative rule (trig) and leaves the inner derivative symbolic; step 4 evaluates the simple inner derivative of 4x+1, which the contract counts as part of a chain-rule step, so both labels are acceptable.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — The math is correct and every line was proved by SymPy. The rule labels (constant-multiple, trig, chain, algebra) each name a rule the step legitimately applied, and the final answer 2*sin(4*x+1) matches.
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 is labeled 'trig' but applies the chain rule (Derivative(cos(u)) = -sin(u)*u'), which is not a trigonometric identity. Step 4 is labeled 'chain' but performs algebraic simplification of the inner derivative, not the chain rule itself. The labels are swapped and incorrect.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'trig' but actually applies the chain rule (differentiating the composite function cos(4x+1)). Step 4 is labeled 'chain' but only differentiates the inner linear function, which is a basic derivative step, not the chain rule itself. The labels for the chain rule application are swapped/misapplied.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'trig' but actually applies the chain rule to differentiate the composite function cos(4x+1). Step 4 is labeled 'chain' but only differentiates the inner linear function, which is a basic derivative step, not the chain rule itself. The labels are swapped and incorrect.
gpt-oss:20b: pass 2026-09-26

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 2026-09-27 with SymPy 1.14.0.