Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1115

Derivative of \( \displaystyle - \frac{\ln{\left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)} \right)}}{4} \)

Problem 2.1115 · hard

Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)} \right)}}{4} \).

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)} \right)}}{4}\right) \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)} \right)}}{4} \]
logarithmicApply the derivative rule for the natural logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}\right)}{4 \left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan{\left(4 x - 1 \right)} + \frac{d}{d x} \sec{\left(4 x - 1 \right)}}{4 \left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}\right)} \]
chainApply the chain rule to both tangent and secant terms.✓ Proved
\[ = - \frac{4 \tan{\left(4 x - 1 \right)} \sec{\left(4 x - 1 \right)} + 4 \sec^{2}{\left(4 x - 1 \right)}}{4 \left(\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}\right)} \]
constant-multipleDifferentiate the inner functions and multiply by the inner derivative 4.≈ Checked numerically
\[ = - \frac{\tan{\left(4 x - 1 \right)} \sec{\left(4 x - 1 \right)} + \sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}} \]
algebraFactor out the 4 from the sum.✓ Proved
\[ = \frac{- \tan{\left(4 x - 1 \right)} \sec{\left(4 x - 1 \right)} - \sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)} + \sec{\left(4 x - 1 \right)}} \]
algebraSimplify the fraction by canceling the 4 and the 1/4.✓ Proved
\[ = - \sec{\left(4 x - 1 \right)} \]
algebra simplifyFactor out sec(4*x - 1) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved

Answer \( - \frac{1}{\cos{\left(4 x - 1 \right)}} \)

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.

✓ 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(4x - 1) + sec(4x - 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(4x - 1) + sec(4x - 1) = 0
5	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(4x - 1)2 + sec(4x - 1)*2 - 1)/(tan(4x - 1) + sec(4x - 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(4x - 1) + sec(4*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(4x - 1) + sec(4x - 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(4x - 1) + sec(4x - 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 tan(4x - 1) + sec(4x - 1) = 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(4*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: fail (style) — Step 4 incorrectly labels the operation as "chain"; it is merely the sum rule applied to the derivatives of tan and sec, not a chain rule application.
qwen3.6:27b-mlx: fail (style) — Step 1 is labeled 'constant-multiple' but performs no differentiation; it is merely a rewrite of the expression. Step 5 is labeled 'constant-multiple' but actually applies the chain rule and standard derivatives of trig functions, which should be labeled 'chain' or 'derivative'.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 1 is labeled 'constant-multiple' but performs no differentiation; it is merely a rewrite of the expression. Step 5 is labeled 'constant-multiple' but actually applies the chain rule and standard derivatives of trig functions, which should be labeled 'chain' or 'derivative'.
gpt-oss:20b: fail (style) 2026-09-28 — Step 4 incorrectly labels the operation as "chain"; it is merely the sum rule applied to the derivatives of tan and sec, not a chain rule application.
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 4 is labeled 'chain' but performs the differentiation of the sum (splitting the derivative), which should be labeled 'sum'. Step 5 is labeled 'constant-multiple' but performs the actual differentiation of tan and sec using the chain rule, which should be labeled 'derivative' or 'chain'. The labels do not match the operations performed.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.