Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.468

Derivative of \( \displaystyle - \frac{\ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)}}{2} \)

Problem 2.468 · hard

Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)}}{2} \).

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)}}{2}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)} \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)}\right)}{2 \left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)}\right)} \]
logarithmicApply the derivative rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan{\left(2 x - 1 \right)} + \frac{d}{d x} \sec{\left(2 x - 1 \right)}}{2 \left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{2 \tan{\left(2 x - 1 \right)} \sec{\left(2 x - 1 \right)} + 2 \sec^{2}{\left(2 x - 1 \right)}}{2 \left(\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)}\right)} \]
chain algebraApply the chain rule to both terms. Rearrange the terms for clarity.≈ Checked numerically
\[ = \frac{- 2 \tan{\left(2 x - 1 \right)} \sec{\left(2 x - 1 \right)} - 2 \sec^{2}{\left(2 x - 1 \right)}}{2 \tan{\left(2 x - 1 \right)} + 2 \sec{\left(2 x - 1 \right)}} \]
algebraCombine the fractions.✓ Proved
\[ = \frac{- \tan{\left(2 x - 1 \right)} \sec{\left(2 x - 1 \right)} - \sec^{2}{\left(2 x - 1 \right)}}{\tan{\left(2 x - 1 \right)} + \sec{\left(2 x - 1 \right)}} \]
simplifyCancel the common factor of 2.✓ Proved
\[ = - \sec{\left(2 x - 1 \right)} \]
algebra simplifyFactor out sec(2*x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved

Answer \( - \frac{1}{\cos{\left(2 x - 1 \right)}} \)

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.

Lines: 10 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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(2x - 1) + sec(2x - 1) = 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(2x - 1) + sec(2x - 1) = 0
5	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(2x - 1)2 + sec(2x - 1)*2 - 1)/(tan(2x - 1) + sec(2x - 1)); 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(2x - 1) + sec(2*x - 1) = 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(2x - 1) + sec(2x - 1) = 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(2x - 1) + sec(2x - 1) = 0 undefined where 2tan(2x - 1) + 2sec(2x - 1) = 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 2tan(2x - 1) + 2sec(2x - 1) = 0 undefined where tan(2x - 1) + sec(2x - 1) = 0
9	✓ 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(2x - 1) + sec(2x - 1) = 0
10	✓ 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(2*x - 1) = 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 — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-21 — 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 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (style) 2026-09-20 — Step 5 applies the chain rule to two separate terms in a single line, violating the rule that each step must change only one thing. The step should be split into two separate chain‑rule applications.
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final result is correct.
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies the chain rule to both the tan and sec terms in a single step, changing two terms at once. Each step should modify only one component, so this step violates the granularity rule.

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.