Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.893

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

Problem 2.893 · medium Video

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

\[ \frac{d}{d x} \left(- \frac{3 \cos{\left(5 x \right)}}{5}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{3 \frac{d}{d x} \cos{\left(5 x \right)}}{5} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{3 \sin{\left(5 x \right)} \frac{d}{d x} 5 x}{5} \]
chainApply the chain rule to the cosine function.✓ Proved
\[ = 3 \sin{\left(5 x \right)} \]
derivative simplifyDifferentiate the inner function 5*x. Simplify the expression by multiplying the constants.✓ Proved

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

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

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The final simplification is algebraically correct and properly labeled.
gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a rule label; it should be marked as "derivative" to indicate the application of the derivative operator to the entire function.
qwen3.6:27b-mlx: pass 2026-09-26
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.