Derivative of \( \displaystyle \ln{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \)

Problem 2.58 · hard Beautiful

Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\tan{\left(x \right)} + \sec{\left(x \right)}\right)}{\tan{\left(x \right)} + \sec{\left(x \right)}} \]
chainApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \tan{\left(x \right)} + \frac{d}{d x} \sec{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = \frac{\tan{\left(x \right)} \sec{\left(x \right)} + \sec^{2}{\left(x \right)}}{\tan{\left(x \right)} + \sec{\left(x \right)}} \]
trigDifferentiate the trigonometric functions.✓ Proved
\[ = \sec{\left(x \right)} \]
algebra simplifyFactor out sec(x) from the numerator. Cancel the common term (tan(x) + sec(x)).✓ Proved

Answer \( \sec{\left(x \right)} \)

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(x) + sec(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(x) + sec(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(x) + sec(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(x) + sec(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 sec has poles at odd multiples of pi/2
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, and standard trigonometric derivatives. The algebraic simplification in the final steps is valid and clearly labeled.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, and standard trigonometric derivatives. The algebraic simplification in the final steps is valid and clearly labeled.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in separate steps. The algebraic simplification is handled in distinct steps, adhering to the one-change-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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.