Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.505

Derivative of \( \displaystyle - \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)} \)

Problem 2.505 · hard

Differentiate \( \displaystyle f(x) = - \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)} \).

\[ \frac{d}{d x} \left(- \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)} \right)} \]
constantMove the constant negative sign outside.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}\right)}{\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan{\left(x - 3 \right)} + \frac{d}{d x} \sec{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{\tan{\left(x - 3 \right)} \sec{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right) + \sec^{2}{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)}{\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}} \]
trigApply the derivatives of tangent and secant.≈ Checked numerically
\[ = - \frac{\tan{\left(x - 3 \right)} \sec{\left(x - 3 \right)} + \sec^{2}{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}} \]
derivativeDifferentiate the inner linear functions.✓ Proved
\[ = \frac{- \tan{\left(x - 3 \right)} \sec{\left(x - 3 \right)} - \sec^{2}{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)} + \sec{\left(x - 3 \right)}} \]
algebraCombine the terms into a single fraction.✓ Proved
\[ = - \sec{\left(x - 3 \right)} \]
algebra simplifyFactor out sec(x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved

Answer \( - \frac{1}{\cos{\left(x - 3 \right)}} \)

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 Lines: 9 proved, 1 checked numerically. 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(x - 3) + sec(x - 3) = 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 - 3) + sec(x - 3) = 0
5	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(x - 3)2 + sec(x - 3)2 - 1)/(tan(x - 3) + sec(x - 3)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(x - 3) + sec(x - 3) = 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(x - 3) + sec(x - 3) = 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(x - 3) + sec(x - 3) = 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(x - 3) + sec(x - 3) = 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(x - 3) = 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 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, and trigonometric derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-20

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.