Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.709

Derivative of \( \displaystyle \frac{5 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \)

Problem 2.709 · hard Video

Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \).

\[ \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]
logarithmicApply the derivative rule for the logarithm.✓ Proved
\[ = \frac{5 \left(\frac{d}{d x} \tan{\left(4 x + 1 \right)} + \frac{d}{d x} \sec{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]
sumApply the sum rule to the inner terms.✓ Proved
\[ = \frac{5 \left(4 \tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + 4 \sec^{2}{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]
chainApply the chain rule to both trigonometric terms.✓ Checked independently
\[ = \frac{5 \left(\tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + \sec^{2}{\left(4 x + 1 \right)}\right)}{\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}} \]
algebraFactor out the common factor of 4.✓ Proved
\[ = \frac{5 \tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + 5 \sec^{2}{\left(4 x + 1 \right)}}{\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}} \]
algebraSimplify the constant coefficients.✓ Proved
\[ = 5 \sec{\left(4 x + 1 \right)} \]
algebra simplifyFactor out sec(4*x + 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved

Answer \( \frac{5}{\cos{\left(4 x + 1 \right)}} \)

Top: the graph of f = 5log(tan(4x + 1) + sec(4x + 1))/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(4tan(4x + 1)² + 4tan(4x + 1)·sec(4x + 1) + 4)/(4(tan(4x + 1) + sec(4x + 1))).

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 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 + 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 independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left 5(tan(4x + 1)*2 - sec(4x + 1)*2 + 1)/(tan(4x + 1) + sec(4x + 1)); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: d/dx tan(u)=4 sec^2 u and d/dx sec(u)=4 sec u tan u verified via rewrite(cos); whole-line difference reduces to 0 (`checks/2.709/line5-db0ae710.py`) 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 5 applies the chain rule to two different terms in a single line, violating the rule that each step must change only one thing. This multi‑rule application should be split into two separate steps.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-12: pass — Each step is mathematically correct and the labels are legitimate. The derivative is 5 sec(4x+1), which equals the stated answer 5/cos(4x+1).

gpt-oss:20b: dismiss — Applying the chain rule to both trig terms is one rule applied uniformly, and the contract counts this as one change; the inner derivative 4 is evaluated as part of the chain step.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — Each step is mathematically correct and the labels are legitimate. The derivative is 5 sec(4x+1), which equals the stated answer 5/cos(4x+1).
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (style) 2026-09-26 — Step 5 applies the chain rule to two different terms in a single line, violating the rule that each step must change only one thing. This multi‑rule application should be split into two separate steps.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the chain rule twice (to tan and to sec) in a single line, violating the rule that each step must change only one thing. The label "chain" is therefore misleading for that step.
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately describe the operations performed at each stage.
gpt-oss:20b: pass 2026-09-21

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.