Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.523

Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \)

Problem 2.523 · hard Video

Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \).

\[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \]
constant-multipleMove the constant factor outside the derivative.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \]
chainApply the chain rule to the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}\right)}{2 \left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}\right)} \]
sumDifferentiate the sum inside the numerator.✓ Proved
\[ = \frac{\frac{\frac{d}{d x} \tan{\left(4 x + 2 \right)}}{2} + \frac{\frac{d}{d x} \sec{\left(4 x + 2 \right)}}{2}}{\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}} \]
chainApply the chain rule to both the tangent and secant terms.✓ Proved
\[ = \frac{2 \tan{\left(4 x + 2 \right)} \sec{\left(4 x + 2 \right)} + 2 \sec^{2}{\left(4 x + 2 \right)}}{\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}} \]
algebra algebraFactor out the common constant 4 from the numerator. Factor out the secant term in the numerator.✓ Checked independently
\[ = 2 \sec{\left(4 x + 2 \right)} \]
simplify simplifyCancel the common factor in the numerator and denominator. Final simplified result.✓ Proved

Answer \( \frac{2}{\cos{\left(4 x + 2 \right)}} \)

Top: the graph of f = log(tan(4x + 2) + sec(4x + 2))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (4tan(4x + 2)² + 4tan(4x + 2)·sec(4x + 2) + 4)/(2(tan(4x + 2) + sec(4x + 2))).

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: 8 proved, 1 checked by independent solution. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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 + 2) + sec(4x + 2) = 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 + 2) + sec(4x + 2) = 0
5	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left 2(tan(4x + 2)*2 - sec(4x + 2)*2 + 1)/(tan(4x + 2) + sec(4x + 2)); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: derivatives evaluated via doit and difference vanishes after rewriting in cos (`checks/2.523/line5-b105a201.py`) tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(4x + 2) + sec(4*x + 2) = 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 + 2) + sec(4x + 2) = 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 + 2) + sec(4x + 2) = 0
8	✓ 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 + 2) = 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: fail (error) — Step 4 applies the chain rule to differentiate the inner functions, but it leaves the derivative operators `Derivative(..., x)` unevaluated. The label 'chain' is insufficient because the step also requires applying the standard derivatives of tan and sec (rule 'derivative') to resolve the operators. A single step cannot combine the structural application of the chain rule with the evaluation of the specific trigonometric derivatives without violating the one-rule-per-step constraint or mislabeling the evaluation part.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The labels describe the step leaving each line: the log chain rule, the sum rule, then the chain rule on tan and sec, then algebra, and each is applied correctly. The derivative is 2*sec(4x+2), which matches the stated answer 2/cos(4x+2).

qwen3.6:27b-mlx: dismiss — Evaluating the derivatives of tan(4x+2) and sec(4x+2) with their inner derivative 4 is part of a chain-rule step, so the 'chain' label is legitimate and the step does one thing.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — The labels describe the step leaving each line: the log chain rule, the sum rule, then the chain rule on tan and sec, then algebra, and each is applied correctly. The derivative is 2*sec(4x+2), which matches the stated answer 2/cos(4x+2).
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 applies the chain rule to differentiate the inner functions, but it leaves the derivative operators `Derivative(..., x)` unevaluated. The label 'chain' is insufficient because the step also requires applying the standard derivatives of tan and sec (rule 'derivative') to resolve the operators. A single step cannot combine the structural application of the chain rule with the evaluation of the specific trigonometric derivatives without violating the one-rule-per-step constraint or mislabeling the evaluation part.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied (e.g., chain rule for composite functions, algebra for factoring and canceling).
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 claims to factor out the constant 4, but the expression is left unchanged. Step 6 performs two operations—factoring out 4 and simplifying the 1/2 factor—in a single step, violating the one‑rule‑per‑step 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.