Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x + 1 \right)} \right)}}{3} \)
Problem 2.1412 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x + 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} + \frac{\ln{\left(\tan{\left(3 x + 1 \right)} \right)}}{3}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6}\right) + \frac{d}{d x} \frac{\ln{\left(\tan{\left(3 x + 1 \right)} \right)}}{3} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} + \frac{\frac{d}{d x} \ln{\left(\tan{\left(3 x + 1 \right)} \right)}}{3} \]constant-multipleFactor out the constants.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x + 1 \right)}}{3 \tan{\left(3 x + 1 \right)}} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)}{6 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x + 1 \right)}}{3 \tan{\left(3 x + 1 \right)}} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(3 x + 1 \right)}}{6 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]sumApply the sum rule inside the first derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x + 1 \right)}}{3 \tan{\left(3 x + 1 \right)}} - \frac{\tan{\left(3 x + 1 \right)} \frac{d}{d x} \tan{\left(3 x + 1 \right)}}{3 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]powerApply the power rule to the squared tangent term.✓ Proved
- \[ = \frac{\sec^{2}{\left(3 x + 1 \right)}}{\tan{\left(3 x + 1 \right)}} - \frac{\tan{\left(3 x + 1 \right)} \sec^{2}{\left(3 x + 1 \right)}}{\tan^{2}{\left(3 x + 1 \right)} + 1} \]chain algebra algebraApply the chain rule to the tangent term. Multiply the terms in the numerator. Simplify the coefficients.≈ Checked numerically
- \[ = \left(\frac{1}{\tan{\left(3 x + 1 \right)}} - \frac{\tan{\left(3 x + 1 \right)}}{\tan^{2}{\left(3 x + 1 \right)} + 1}\right) \sec^{2}{\left(3 x + 1 \right)} \]algebraFactor out the common secant term.✓ Proved
- \[ = \frac{\sec^{2}{\left(3 x + 1 \right)}}{\left(\tan^{2}{\left(3 x + 1 \right)} + 1\right) \tan{\left(3 x + 1 \right)}} \]algebra algebra algebraCombine the fractions using a common denominator. Simplify the numerator by canceling the tangent squared terms. Rewrite the expression as a single fraction.✓ Proved
- \[ = \frac{1}{\tan{\left(3 x + 1 \right)}} \]algebra algebra algebraSubstitute the identity 1 = sec^2(u) - tan^2(u) is not needed; instead, use 1 + tan^2 = sec^2. Use the identity 1 + tan(3*x + 1)**2 = sec(3*x + 1)**2. Cancel the secant squared term.≈ Checked numerically
- \[ = \cot{\left(3 x + 1 \right)} \]simplifyFinal simplification to cotangent.✓ Proved
Answer \( \frac{1}{\tan{\left(3 x + 1 \right)}} \)
✓ Nihil obstat Lines: 16 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |
| 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 |
| 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 tan(3*x + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 5 | ✓ 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 + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 6 | ✓ 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 + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(3*x + 1)**2 - sec(3*x + 1)**2 + 1)/(tan(3*x + 1)**3 + tan(3*x + 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(3*x + 1)**2 + 1 = 0 undefined where tan(3*x + 1) = 0 sec has poles at odd multiples of pi/2 |
| 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(3*x + 1)**2 + 1 = 0 undefined where tan(3*x + 1) = 0 |
| 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(3*x + 1)**2 + 1 = 0 undefined where tan(3*x + 1) = 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(3*x + 1)**2 + 1 = 0 undefined where tan(3*x + 1) = 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 + 1)**2 + 1 = 0 undefined where tan(3*x + 1) = 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 + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 13 | ✓ 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 + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 14 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-tan(3*x + 1)**2 + sec(3*x + 1)**2 - 1)/(tan(3*x + 1)**3 + tan(3*x + 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(3*x + 1) = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| 15 | ✓ 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 + 1) = 0 |
| 16 | ✓ 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 + 1) = 0 |
| 17 | ✓ 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 + 1) = 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 + 1) = 0 |
| 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: fail (style) — Step 1 is labeled 'derivative', but the definition restricts 'derivative' to unfolding d/dx on a known form (like d/dx(x) or d/dx(sin(x))). Step 1 is merely the initial setup of the problem, not the application of a differentiation rule to a specific term. It should be labeled 'rewrite' or omitted as a trivial step, but labeling it 'derivative' violates the contract's definition of that label.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 1 is labeled 'derivative', but the definition restricts 'derivative' to unfolding d/dx on a known form (like d/dx(x) or d/dx(sin(x))). Step 1 is merely the initial setup of the problem, not the application of a differentiation rule to a specific term. It should be labeled 'rewrite' or omitted as a trivial step, but labeling it 'derivative' violates the contract's definition of that label.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the one-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03
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.