Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1256

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

Problem 2.1256 · hard Video

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

\[ \frac{d}{d x} \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{3} \]
constant-multipleStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{3} \]
logarithmicPull out the constant factor.✓ Proved
\[ = \frac{\frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(3 x - 3 \right)} \right)}\right)}{3} \]
logarithmicUse the logarithm power rule to bring down the exponent.✓ Proved
\[ = - \frac{2 \frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)}}{3} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{2 \frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{2 \sin{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \cos{\left(3 x - 3 \right)}} \]
chainApply the chain rule for the cosine function.✓ Proved
\[ = \frac{2 \sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} \]
derivative algebraDifferentiate the innermost linear function. Simplify the constants and signs.✓ Proved
\[ = 2 \tan{\left(3 x - 3 \right)} \]
trigUse the identity sin(u)/cos(u) = tan(u).✓ Proved

Answer \( 2 \tan{\left(3 x - 3 \right)} \)

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

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 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. 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 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 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
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(3*x - 3) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(3*x - 3) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(3*x - 3) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(3*x - 3) = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(3*x - 3) = 0 tan 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 tan has poles at odd multiples of pi/2
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 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the constant multiple rule, logarithmic power rule, and chain rule in separate steps. All labels are appropriate and the logic is sound.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 2 is labeled 'logarithmic' but performs the 'constant-multiple' rule (pulling out the 1/3 factor). Step 3 is labeled 'logarithmic' but performs an algebraic rewrite of the log argument. The labels do not match the operations performed.
gpt-oss:20b: fail (style) 2026-09-29 — Step 2 incorrectly labels the operation as "logarithmic"; it is a constant‑multiple step that pulls the factor 1/3 out of the derivative.

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 2026-10-03 with SymPy 1.14.0.