Derivative of \( \displaystyle \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} \)
Problem 2.1215 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(3 x - 3 \right)}} \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(3 x - 3 \right)} \right)}\right)}{6} \]algebraUse the logarithm power rule to simplify the expression inside the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(3 x - 3 \right)} \right)}}{3} \]constant-multiple algebraPull out the constant factor from the derivative. Simplify the constant coefficient.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(3 x - 3 \right)}}{3 \cos{\left(3 x - 3 \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\sin{\left(3 x - 3 \right)} \frac{d}{d x} \left(3 x - 3\right)}{3 \cos{\left(3 x - 3 \right)}} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = \frac{\sin{\left(3 x - 3 \right)}}{\cos{\left(3 x - 3 \right)}} \]derivative algebra algebraDifferentiate the innermost linear function. Simplify the constant coefficients. Simplify the expression.✓ Proved
- \[ = \tan{\left(3 x - 3 \right)} \]trigUse the tangent identity.✓ Proved
Answer \( \tan{\left(3 x - 3 \right)} \)
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.
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.
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 |
| 6 | ✓ 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 |
| 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x - 3) = 0 |
| 11 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the rules in a step-by-step manner, adhering to the one-change-per-step constraint. The use of logarithm properties to simplify the expression before differentiating is valid and clearly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the rules in a step-by-step manner, adhering to the one-change-per-step constraint. The use of logarithm properties to simplify the expression before differentiating is valid and clearly labeled.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the constant multiple rule, logarithm power rule, and chain rule in separate steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 incorrectly rewrites log(cos(3*x - 3)**(-2)) as -2*log(cos(3*x - 3)). The logarithm power rule requires the argument to be positive; cos(3*x - 3) can be negative, so this transformation is not universally valid. The step applies more than one rule (logarithm power rule and algebraic simplification) in one line, violating the single‑rule‑per‑step rule.
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 2026-09-29 with SymPy 1.14.0.