Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.578

Derivative of \( \displaystyle - \frac{\cos{\left(4 x - 3 \right)}}{4} \)

Problem 2.578 · medium Video

Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x - 3 \right)}}{4} \).

\[ \frac{d}{d x} \left(- \frac{\cos{\left(4 x - 3 \right)}}{4}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \cos{\left(4 x - 3 \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{\sin{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right)}{4} \]
trigApply the derivative rule for cosine.✓ Proved
\[ = \sin{\left(4 x - 3 \right)} \]
derivative simplifyDifferentiate the inner function 4*x - 3. Simplify the resulting expression.✓ Proved

Answer \( \sin{\left(4 x - 3 \right)} \)

Top: the graph of f = -cos(4x - 3)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sin(4x - 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: pass

Senior review claude-sonnet-5-5, 2026-10-10: pass — The derivation is correct: -1/4 times the derivative of cos(4x-3) gives -1/4 · (-sin(4x-3) · 4) = sin(4x-3). Each step changes one thing and its rule label is legitimate. SymPy proved every line and the final answer. The panel listed no specific objection to rule on.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — The derivation is correct: -1/4 times the derivative of cos(4x-3) gives -1/4 · (-sin(4x-3) · 4) = sin(4x-3). Each step changes one thing and its rule label is legitimate. SymPy proved every line and the final answer. The panel listed no specific objection to rule on.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies both the cosine derivative rule and the chain rule in one line, but only labels it as "trig". This violates the rule that each step must change only one thing and must label the correct rule(s).
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule (labeled as trig/derivative steps), and simplification. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as "trig"; the correct label should be "chain" (or "product" if treating the product of -sin and the inner 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.