Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.2474

Derivative of \( \displaystyle \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} - \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \)

Problem 2.2474 · hard Video

Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} - \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \).

\[ \frac{d}{d x} \left(\frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} - \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} - \frac{d}{d x} \frac{\ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]
sumApply the sum rule.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]
constant-multipleFactor out the constants.✓ Proved
\[ = \frac{\frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(3 x - 3 \right)} \right)}\right)}{6} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]
logarithmicUse the property log(a^b) = b*log(a) to simplify the first term.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)}}{3} - \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x - 3 \right)} \right)}}{3} \]
constant-multiple algebraFactor out the -2. Simplify the coefficient -2/6.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan{\left(3 x - 3 \right)}}{3 \tan{\left(3 x - 3 \right)}} - \frac{\frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]
chainApply the chain rule to both logarithmic terms.✓ Proved
\[ = \frac{\sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} - \frac{\sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]
chain algebraDifferentiate the inner functions. Simplify the coefficients and products.✓ Proved
\[ = \tan{\left(3 x - 3 \right)} - \frac{\sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]
algebraSubstitute tan(u) = sin(u)/cos(u).✓ Proved
\[ = \frac{\tan^{2}{\left(3 x - 3 \right)} - \sec^{2}{\left(3 x - 3 \right)}}{\tan{\left(3 x - 3 \right)}} \]
algebraCombine the terms over a common denominator.✓ Proved
\[ = - \frac{1}{\tan{\left(3 x - 3 \right)}} \]
trig algebraUse the identity sec^2(u) = 1 + tan^2(u). Simplify the numerator.✓ Proved
\[ = - \cot{\left(3 x - 3 \right)} \]
trigUse the identity cot(u) = 1/tan(u).✓ Proved

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

Top: the graph of f = log(cos(3x - 3)^(-2))/6 - log(tan(3x - 3))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(3tan(3x - 3)² + 3)/(3tan(3x - 3)) + sin(3x - 3)/cos(3x - 3).

Every line of this solution was proved by the computer algebra system SymPy. 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 undefined where cos(3*x - 3) = 0
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 undefined where cos(3*x - 3) = 0
4	✓ 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 undefined where cos(3*x - 3) = 0
5	✓ 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
6	✓ 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
7	✓ 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 undefined where tan(3x - 3) = 0 undefined where cos(3x - 3) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(3x - 3) = 0 undefined where cos(3x - 3) = 0 sec has poles at odd multiples of pi/2
9	✓ 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 - 3) = 0 undefined where cos(3x - 3) = 0
10	✓ 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 - 3) = 0 undefined where cos(3x - 3) = 0
11	✓ 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(3*x - 3) = 0
12	✓ 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(3*x - 3) = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(3*x - 3) = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(3*x - 3) = 0 cot has poles at multiples of pi
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(3*x - 3) = 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-23
gpt-oss:20b: pass 2026-10-23
qwen3.6:27b-mlx: pass 2026-10-22
gpt-oss:20b: pass 2026-10-22
qwen3.6:27b-mlx: fail (error) 2026-10-22 — Step 7 applies the chain rule to both terms simultaneously, violating the constraint that each step must change only one thing. Additionally, Step 8 is labeled 'chain' but actually performs the differentiation of the inner trigonometric functions (derivative rule), not the chain rule application itself.
gpt-oss:20b: pass 2026-10-22

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 2026-10-23 with SymPy 1.14.0.