Derivative of \( \displaystyle - \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(3 x + 2 \right)} \right)} \)
Problem 2.704 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(3 x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2} + \ln{\left(\tan{\left(3 x + 2 \right)} \right)}\right) \]sumDifferentiate the sum of two terms.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} \]constantApply the constant multiple rule to the first term.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(3 x + 2 \right)} + 1 \right)}}{2} + \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)}{2 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]sumDifferentiate the inner sum.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(3 x + 2 \right)}}{2 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]constantDifferentiate the constant term 1.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} - \frac{\frac{d}{d x} \tan^{2}{\left(3 x + 2 \right)}}{2 \left(\tan^{2}{\left(3 x + 2 \right)} + 1\right)} \]powerApply the power rule to the squared tangent term.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} - \frac{\tan{\left(3 x + 2 \right)} \frac{d}{d x} \tan{\left(3 x + 2 \right)}}{\tan^{2}{\left(3 x + 2 \right)} + 1} \]chainApply the chain rule to the squared tangent term.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(3 x + 2 \right)} \right)} - \frac{3 \tan{\left(3 x + 2 \right)} \sec^{2}{\left(3 x + 2 \right)}}{\tan^{2}{\left(3 x + 2 \right)} + 1} \]chain algebraApply the chain rule to the tangent function. Multiply the constants 2 and 3.≈ Checked numerically
- \[ = \frac{3 \sec^{2}{\left(3 x + 2 \right)}}{\tan{\left(3 x + 2 \right)}} - \frac{3 \tan{\left(3 x + 2 \right)}}{\left(\tan^{2}{\left(3 x + 2 \right)} + 1\right) \cos^{2}{\left(3 x + 2 \right)}} \]chain algebraApply the chain rule to the second logarithm term. Simplify the expression by distributing -1/2 and combining terms.≈ Checked numerically
- \[ = \frac{3}{\cos^{2}{\left(3 x + 2 \right)} \tan{\left(3 x + 2 \right)}} - \frac{3 \tan{\left(3 x + 2 \right)}}{\left(\tan^{2}{\left(3 x + 2 \right)} + 1\right) \cos^{2}{\left(3 x + 2 \right)}} \]algebraRewrite sec(x)**2 as 1/cos(x)**2.✓ Proved
- \[ = \frac{3}{\cos^{2}{\left(3 x + 2 \right)} \tan{\left(3 x + 2 \right)}} - \frac{3 \tan{\left(3 x + 2 \right)}}{\left(\frac{\sin^{2}{\left(3 x + 2 \right)}}{\cos^{2}{\left(3 x + 2 \right)}} + 1\right) \cos^{2}{\left(3 x + 2 \right)}} \]algebraSubstitute tan(x)**2 with sin(x)**2/cos(x)**2.✓ Proved
- \[ = - 3 \tan{\left(3 x + 2 \right)} + \frac{3}{\cos^{2}{\left(3 x + 2 \right)} \tan{\left(3 x + 2 \right)}} \]algebra algebraSimplify the denominator of the first term. Cancel out the cos(3*x + 2)**2 terms in the first fraction.✓ Proved
- \[ = - 3 \tan{\left(3 x + 2 \right)} + \frac{3}{\sin{\left(3 x + 2 \right)} \cos{\left(3 x + 2 \right)}} \]algebra algebra algebraRewrite 1/tan(x) as 1/sin(x) * cos(x). Simplify the second term by canceling one cos(3*x + 2). Introduce a factor of 2 in the denominator to prepare for the double angle identity.✓ Proved
- \[ = - 3 \tan{\left(3 x + 2 \right)} + \frac{6}{\sin{\left(6 x + 4 \right)}} \]algebra algebraUse the double angle identity sin(2u) = 2sin(u)cos(u). Distribute the 2 inside the sine function.✓ Proved
- \[ = - 3 \tan{\left(3 x + 2 \right)} + 6 \csc{\left(6 x + 4 \right)} \]rewrite simplifyRewrite 1/sin(x) as csc(x). Rearrange the terms into a standard form.✓ Proved
Answer \( \frac{3}{\tan{\left(3 x + 2 \right)}} \)
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: 20 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 |
| 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 + 2)**2 + 1 = 0 |
| 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 undefined where tan(3*x + 2)**2 + 1 = 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 undefined where tan(3*x + 2)**2 + 1 = 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 undefined where tan(3*x + 2)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*(-tan(3*x + 2)**2 + sec(3*x + 2)**2 - 1)*tan(3*x + 2)/(tan(3*x + 2)**2 + 1); 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(3*x + 2)**2 + 1 = 0 sec has poles at odd multiples of pi/2 |
| 9 | ✓ 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(3*x + 2)**2 + 1 = 0 |
| 10 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*((tan(3*x + 2)**2 + 1)*(tan(3*x + 2)**2 - sec(3*x + 2)**2 + 1)*cos(3*x + 2)**2 - sin(3*x + 2)**2/cos(3*x + 2)**2 + tan(3*x + 2)**2)/((tan(3*x + 2)**2 + 1)*cos(3*x + 2)**2*tan(3*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 sec has poles at odd multiples of pi/2 undefined where tan(3*x + 2)**2 + 1 = 0 undefined where tan(3*x + 2) = 0 undefined where cos(3*x + 2) = 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 + 2) = 0 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2)**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(3*x + 2) = 0 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2)**2 + 1 = 0 |
| 13 | ✓ 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 + 2)**2 + 1 = 0 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2) = 0 undefined where sin(3*x + 2)**2/cos(3*x + 2)**2 + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x + 2)**2/cos(3*x + 2)**2 + 1 = 0 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(3*x + 2) = 0 undefined where tan(3*x + 2) = 0 undefined where sin(3*x + 2) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x + 2) = 0 undefined where cos(3*x + 2) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x + 2) = 0 undefined where cos(3*x + 2) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(3*x + 2) = 0 undefined where cos(3*x + 2) = 0 undefined where sin(6*x + 4) = 0 |
| 20 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(6*x + 4) = 0 |
| 21 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(6*x + 4) = 0 csc has poles at multiples of pi |
| 22 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 csc has poles at multiples of pi |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -3*tan(3*x + 2) + 6*csc(6*x + 4) - 3/tan(3*x + 2); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(3*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: fail (error) — The solution incorrectly simplifies the term 3/(sin(3x+2)*cos(3x+2)) to 6/sin(6x+4). This step applies two rules at once (double‑angle identity and algebraic manipulation) and yields a value that is not equal to the original expression. The correct derivative is 3/tan(3x+2).qwen3.6:27b-mlx: fail (error) — The final answer is mathematically incorrect; the derivative of -log(tan^2+1)/2 simplifies to -tan(3x+2), which does not cancel with the second term to yield 3/tan(3x+2). Additionally, Step 2 is labeled 'constant' but performs the linearity of differentiation (splitting the sum), which should be labeled 'sum' or handled in Step 1, while Step 3 is labeled 'chain' but performs the constant multiple rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — The final answer is mathematically incorrect; the derivative of -log(tan^2+1)/2 simplifies to -tan(3x+2), which does not cancel with the second term to yield 3/tan(3x+2). Additionally, Step 2 is labeled 'constant' but performs the linearity of differentiation (splitting the sum), which should be labeled 'sum' or handled in Step 1, while Step 3 is labeled 'chain' but performs the constant multiple rule.gpt-oss:20b: fail (error) 2026-09-21 — The solution incorrectly simplifies the term 3/(sin(3x+2)*cos(3x+2)) to 6/sin(6x+4). This step applies two rules at once (double‑angle identity and algebraic manipulation) and yields a value that is not equal to the original expression. The correct derivative is 3/tan(3x+2).qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 is labeled 'constant' but performs the differentiation of the sum (splitting the derivative), which should be labeled 'sum'. Step 3 is labeled 'chain' but performs the constant multiple rule (pulling out -1/2), which should be labeled 'constant-multiple'. The labels do not match the operations performed.gpt-oss:20b: fail (error) 2026-09-21 — Step 16 incorrectly rewrites 1/tan(3*x+2) as sin(3*x+2)/cos(3*x+2) instead of cos(3*x+2)/sin(3*x+2), leading to an incorrect simplification and ultimately an incorrect final 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 2026-09-26 with SymPy 1.14.0.