Derivative of \( \displaystyle \frac{3 \tan{\left(5 x \right)}}{5} \)
Problem 2.708 · easy
Differentiate \( \displaystyle f(x) = \frac{3 \tan{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \frac{3 \tan{\left(5 x \right)}}{5} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(5 x \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \sec^{2}{\left(5 x \right)} \frac{d}{d x} 5 x}{5} \]chainApply the chain rule to the tangent function.✓ Proved
- \[ = 3 \sec^{2}{\left(5 x \right)} \]derivative algebraDifferentiate the inner function 5*x. Simplify the expression by canceling the 5s.✓ Proved
Answer \( \frac{3}{\cos^{2}{\left(5 x \right)}} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
✓ 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.
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 tan has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(5*x) = 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 is labeled "unlabelled" even though it applies the derivative rule. All steps must name a rule from the allowed vocabulary; the first step should be labeled "derivative".qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative of the inner function in separate steps. The final simplification is algebraic and correctly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and derivative of the inner function in separate steps. The final simplification is algebraic and correctly labeled.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 is labeled "unlabelled" even though it applies the derivative rule. All steps must name a rule from the allowed vocabulary; the first step should be labeled "derivative".qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.