Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.2300

Derivative of \( \displaystyle \frac{2 \ln{\left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)} \right)}}{3} \)

Problem 2.2300 · hard Video

Differentiate \( \displaystyle f(x) = \frac{2 \ln{\left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)} \right)}}{3} \).

\[ \frac{d}{d x} \frac{2 \ln{\left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)} \right)}}{3} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{2 \frac{d}{d x} \ln{\left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)} \right)}}{3} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{2 \frac{d}{d x} \left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}\right)}{3 \left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}\right)} \]
logarithmicApply the chain rule for the logarithm.✓ Proved
\[ = \frac{2 \left(\frac{d}{d x} \tan{\left(3 x - 1 \right)} + \frac{d}{d x} \sec{\left(3 x - 1 \right)}\right)}{3 \left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \frac{2 \left(3 \tan{\left(3 x - 1 \right)} \sec{\left(3 x - 1 \right)} + \frac{d}{d x} \tan{\left(3 x - 1 \right)}\right)}{3 \left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}\right)} \]
chainApply the chain rule to the secant term.✓ Proved
\[ = \frac{2 \left(3 \tan{\left(3 x - 1 \right)} \sec{\left(3 x - 1 \right)} + 3 \sec^{2}{\left(3 x - 1 \right)}\right)}{3 \left(\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}\right)} \]
chainApply the chain rule to the tangent term.≈ Checked numerically
\[ = \frac{2 \tan{\left(3 x - 1 \right)} \sec{\left(3 x - 1 \right)} + 2 \sec^{2}{\left(3 x - 1 \right)}}{\tan{\left(3 x - 1 \right)} + \sec{\left(3 x - 1 \right)}} \]
algebraDistribute the constant and simplify the fraction.✓ Proved
\[ = 2 \sec{\left(3 x - 1 \right)} \]
algebra simplifyFactor out the common term sec(3*x - 1). Cancel the common term in the numerator and denominator.✓ Proved

Answer \( \frac{2}{\cos{\left(3 x - 1 \right)}} \)

Top: the graph of f = 2log(tan(3x - 1) + sec(3x - 1))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2(3tan(3x - 1)² + 3tan(3x - 1)·sec(3x - 1) + 3)/(3(tan(3x - 1) + sec(3x - 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.

Lines: 9 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. 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(3x - 1) + sec(3x - 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(3x - 1) + sec(3x - 1) = 0
5	✓ 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(3x - 1) + sec(3x - 1) = 0
6	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left 2(tan(3x - 1)*2 - sec(3x - 1)*2 + 1)/(tan(3x - 1) + sec(3x - 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(3x - 1) + sec(3*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(3x - 1) + sec(3x - 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(3x - 1) + sec(3x - 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(3*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 (error) — Step 9 incorrectly cancels the common factor, assuming tan(3*x-1)+sec(3*x-1)=sec(3*x-1). This identity is false, so the final result 2*sec(3*x-1) is not the derivative of the given function.
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — 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) 2026-10-17 — Step 9 incorrectly cancels the common factor, assuming tan(3*x-1)+sec(3*x-1)=sec(3*x-1). This identity is false, so the final result 2*sec(3*x-1) is not the derivative of the given function.
qwen3.6:27b-mlx: pass 2026-10-15 — 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: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14

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 2026-10-17 with SymPy 1.14.0.