Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \)
Problem 2.1506 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]constant-multipleMove the constant outside the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]sumDifferentiate the sum inside the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(5 x \right)} + \frac{d}{d x} \sec{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 5 \sec^{2}{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]constant-multipleFactor out the constant 5.✓ Proved
- \[ = \frac{\tan{\left(5 x \right)} \sec{\left(5 x \right)} + \sec^{2}{\left(5 x \right)}}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]algebra simplifySimplify the expression by canceling 5 and 1/5. Simplify the fraction.✓ Proved
- \[ = \sec{\left(5 x \right)} \]algebra simplifyFactor out sec(5*x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(5 x \right)}} \)
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 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 |
| 3 | ✓ 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(5*x) + sec(5*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(5*x) + sec(5*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(5*x) + sec(5*x) = 0 |
| 6 | ✓ 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(5*x) + sec(5*x) = 0 |
| 7 | ✓ 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(5*x) + sec(5*x) = 0 |
| 8 | ✓ 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(5*x) + sec(5*x) = 0 |
| 9 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately describe the operations performed.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (style) 2026-10-04 — Step 3 incorrectly labels the application of the chain rule as "sum"; the derivative of log(u) is 1/u·u', so the correct label is "chain". Step 4 labels the application of the sum rule as "chain"; the derivative of tan(5x)+sec(5x) is the sum of the derivatives, so the correct label is "sum".
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 2026-10-04 with SymPy 1.14.0.