Derivative of \( \displaystyle \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)} \)
Problem 2.910 · hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)}\right)}{\tan{\left(3 x \right)} + \sec{\left(3 x \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x \right)} + \frac{d}{d x} \sec{\left(3 x \right)}}{\tan{\left(3 x \right)} + \sec{\left(3 x \right)}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{3 \tan{\left(3 x \right)} \sec{\left(3 x \right)} + 3 \sec^{2}{\left(3 x \right)}}{\tan{\left(3 x \right)} + \sec{\left(3 x \right)}} \]chainApply the chain rule to each trigonometric term.✓ Proved
- \[ = 3 \sec{\left(3 x \right)} \]algebra simplifyFactor out the common term 3*sec(3*x). Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\cos{\left(3 x \right)}} \)
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 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(3*x) + sec(3*x) = 0 |
| 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 undefined where tan(3*x) + sec(3*x) = 0 |
| 4 | ✓ 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 undefined where tan(3*x) + sec(3*x) = 0 |
| 5 | ✓ 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 undefined where tan(3*x) + sec(3*x) = 0 |
| 6 | ✓ 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(3*x) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, and standard derivatives. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule, sum rule, and standard derivatives. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-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 2026-09-26 with SymPy 1.14.0.