Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x + 2 \right)} \right)} \)
Problem 2.1226 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x + 2 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \frac{d}{d x} \ln{\left(\tan{\left(x + 2 \right)} \right)} \]sumApply the difference rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \frac{\frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} \]logarithmicApply the chain rule for the second term.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \frac{\sec^{2}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)}{\tan{\left(x + 2 \right)}} \]trigDifferentiate the tangent function.≈ Checked numerically
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the expression.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} + \frac{\frac{d}{d x} \left(\tan^{2}{\left(x + 2 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]logarithmicApply the chain rule to the first term.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} + \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x + 2 \right)}}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]derivativeDifferentiate the sum inside the derivative.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} + \frac{\frac{d}{d x} \tan^{2}{\left(x + 2 \right)}}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} + \frac{\tan{\left(x + 2 \right)} \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]powerApply the power rule and chain rule.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)}} + \frac{\tan{\left(x + 2 \right)} \sec^{2}{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]trig algebraDifferentiate the tangent function again. Simplify the coefficients.≈ Checked numerically
- \[ = \left(- \frac{1}{\tan{\left(x + 2 \right)}} + \frac{\tan{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1}\right) \sec^{2}{\left(x + 2 \right)} \]algebraFactor out the common secant term.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\left(\tan^{2}{\left(x + 2 \right)} + 1\right) \tan{\left(x + 2 \right)}} \]algebra algebra algebraFind a common denominator. Simplify the numerator. Final simplification.✓ Proved
- \[ = - \frac{\sec^{2}{\left(x + 2 \right)}}{\tan^{3}{\left(x + 2 \right)} + \tan{\left(x + 2 \right)}} \]algebraExpand the denominator.✓ Proved
Answer \( - \frac{1}{\tan{\left(x + 2 \right)}} \)
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.
Lines: 15 proved, 3 checked numerically. 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 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 undefined where tan(x + 2) = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-tan(x + 2)**2 + sec(x + 2)**2 - 1)/tan(x + 2); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(x + 2) = 0 sec has poles at odd multiples of pi/2 |
| 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 sec has poles at odd multiples of pi/2 undefined where tan(x + 2) = 0 |
| 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 sec has poles at odd multiples of pi/2 undefined where tan(x + 2) = 0 |
| 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 sec has poles at odd multiples of pi/2 undefined where tan(x + 2) = 0 undefined where tan(x + 2)**2 + 1 = 0 |
| 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(x + 2)**2 + 1 = 0 undefined where tan(x + 2) = 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(x + 2)**2 + 1 = 0 undefined where tan(x + 2) = 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 0 |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(x + 2)**2 - sec(x + 2)**2 + 1)*tan(x + 2)/(tan(x + 2)**2 + 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 0 |
| 15 | ✓ 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 0 |
| 16 | ✓ 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 0 |
| 17 | ✓ 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 + 2)**2 + 1 = 0 undefined where tan(x + 2) = 0 undefined where tan(x + 2)**3 + tan(x + 2) = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left (tan(x + 2)**2 - sec(x + 2)**2 + 1)/(tan(x + 2)**3 + tan(x + 2)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x + 2) = 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 10 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 2 labels the application of linearity (constant multiple and difference) as 'sum', which is imprecise given the available vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 10 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 2 labels the application of linearity (constant multiple and difference) as 'sum', which is imprecise given the available vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 10 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, the final answer in step 17 does not match the stated answer of -1/tan(x + 2); the solution fails to simplify sec^2(u)/(tan(u)(tan^2(u)+1)) to 1/tan(u) using the identity tan^2(u)+1 = sec^2(u).gpt-oss:20b: fail (error) 2026-09-29 — Step 10 applies both the power rule and the chain rule in a single step, violating the rule that each step must change only one thing. The label "power" also fails to acknowledge the chain rule applied to tan(x+2).
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.