Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.852

Derivative of \( \displaystyle 3 \ln{\left(\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)} \right)} \)

Problem 2.852 · hard Beautiful Video

Differentiate \( \displaystyle f(x) = 3 \ln{\left(\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)} \right)} \).

\[ \frac{d}{d x} 3 \ln{\left(\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 3 \frac{d}{d x} \ln{\left(\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)} \right)} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)}\right)}{\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{3 \frac{d}{d x} \tan{\left(x + 1 \right)} + 3 \frac{d}{d x} \sec{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)}} \]
sumApply the sum rule to the numerator.✓ Proved
\[ = \frac{3 \tan{\left(x + 1 \right)} \sec{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + 3 \sec^{2}{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)}{\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)}} \]
trigDifferentiate the trigonometric functions.✓ Checked independently
\[ = \frac{3 \tan{\left(x + 1 \right)} \sec{\left(x + 1 \right)} + 3 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)} + \sec{\left(x + 1 \right)}} \]
chain algebraApply the chain rule to the inner function x + 1. Simplify the numerator by multiplying out the 1s.✓ Proved
\[ = 3 \sec{\left(x + 1 \right)} \]
algebra simplifyFactor out the common term sec(x + 1) in the numerator. Cancel the common factor in the numerator and denominator.✓ Proved

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

Top: the graph of f = 3log(tan(x + 1) + sec(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3(tan(x + 1)² + tan(x + 1)·sec(x + 1) + 1)/(tan(x + 1) + sec(x + 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(x + 1) + sec(x + 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(x + 1) + sec(x + 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 3(tan(x + 1)2 - sec(x + 1)*2 + 1)/(tan(x + 1) + sec(x + 1)); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: after doit and rewriting tan/sec in terms of cos, difference simplifies to 0 (uses tan^2+1 = sec^2) (`checks/2.852/line5-baa510b3.py`) tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(x + 1) + sec(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(x + 1) + sec(x + 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(x + 1) + sec(x + 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(x + 1) + sec(x + 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(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 the labels accurately reflect the operations performed.

Senior review claude-sonnet-5-5, 2026-10-13: pass — Every step is correct and uses an allowed rule name. Line 5 differentiates tan and sec in one trig step, which counts as one change. The final line 3*sec(x+1) equals the stated answer 3/cos(x+1).

agent:pipeline: uphold — Beautiful nomination: The log of tan+sec looks messy, but the derivative collapses to 3*sec(x+1) when the numerator factors and cancels against the denominator.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — Every step is correct and uses an allowed rule name. Line 5 differentiates tan and sec in one trig step, which counts as one change. The final line 3*sec(x+1) equals the stated answer 3/cos(x+1).
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.
gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies both the trigonometric derivative rule and the chain rule at once, violating the one‑rule‑per‑step requirement.

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 2026-09-27 with SymPy 1.14.0.