Derivative of \( \displaystyle - \frac{3 \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} + 3 \ln{\left(\tan{\left(x - 1 \right)} \right)} \)
Problem 2.1352 · hard Beautiful
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} + 3 \ln{\left(\tan{\left(x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} + 3 \ln{\left(\tan{\left(x - 1 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} 3 \ln{\left(\tan{\left(x - 1 \right)} \right)} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} + 3 \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} \]constant-multipleFactor out the constants.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{3 \frac{d}{d x} \left(\tan^{2}{\left(x - 1 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]logarithmicApply the derivative rule for logarithms.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{3 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x - 1 \right)}\right)}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]sumApply the sum rule inside the derivative.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{3 \frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{3 \tan{\left(x - 1 \right)} \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]powerApply the power rule and chain rule.✓ Proved
- \[ = \frac{3 \sec^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\tan{\left(x - 1 \right)}} - \frac{3 \tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\tan^{2}{\left(x - 1 \right)} + 1} \]trigApply the derivative rule for tangent.≈ Checked numerically
- \[ = \frac{3 \sec^{2}{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{3 \tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]derivative algebra algebraThe derivative of x - 1 is 1. Simplify the expression. Combine terms into common fractions.✓ Proved
- \[ = 3 \left(\frac{1}{\tan{\left(x - 1 \right)}} - \frac{\tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1}\right) \sec^{2}{\left(x - 1 \right)} \]algebraFactor out the common term 3 * sec(x - 1)**2.✓ Proved
- \[ = \frac{3 \sec^{2}{\left(x - 1 \right)}}{\left(\tan^{2}{\left(x - 1 \right)} + 1\right) \tan{\left(x - 1 \right)}} \]algebra algebraFind a common denominator for the terms inside the parentheses. Simplify the numerator.✓ Proved
- \[ = \frac{3}{\tan{\left(x - 1 \right)}} \]algebra algebraUse the identity tan(x - 1)**2 + 1 = sec(x - 1)**2. Cancel the sec(x - 1)**2 terms.≈ Checked numerically
- \[ = 3 \cot{\left(x - 1 \right)} \]simplifyUse the identity 1/tan(x) = cot(x).✓ Proved
Answer \( \frac{3}{\tan{\left(x - 1 \right)}} \)
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.
✓ 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(x - 1) = 0 undefined where tan(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(x - 1) = 0 undefined where tan(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(x - 1) = 0 undefined where tan(x - 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1) = 0 undefined where tan(x - 1)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (3*tan(x - 1)**2 - 3*sec(x - 1)**2 + 3)/(tan(x - 1)**3 + tan(x - 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x - 1) = 0 undefined where tan(x - 1)**2 + 1 = 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(x - 1) = 0 undefined where tan(x - 1)**2 + 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(x - 1) = 0 undefined where tan(x - 1)**2 + 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(x - 1) = 0 undefined where tan(x - 1)**2 + 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(x - 1) = 0 undefined where tan(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(x - 1) = 0 undefined where tan(x - 1)**2 + 1 = 0 |
| 14 | ✓ 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(x - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0 |
| 15 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-3*tan(x - 1)**2 + 3*sec(x - 1)**2 - 3)/(tan(x - 1)**3 + tan(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(x - 1)**2 + 1 = 0 undefined where tan(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(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(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(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 7 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. Step 8 also applies the chain rule implicitly alongside the trig derivative rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 7 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. Step 8 also applies the chain rule implicitly alongside the trig derivative rule.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-30 — Step 7 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. The label 'power' is also insufficient as it does not account for the chain rule application.gpt-oss:20b: pass 2026-09-30
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.