Derivative of \( \displaystyle - \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{5 \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \)
Problem 2.226 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{5 \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{5 \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4}\right) \]sumDifferentiate the sum of two terms.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8}\right) + \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \]constant-multiplePull out the constant coefficients.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} + \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(4 x - 1 \right)} \right)}}{4} \]constant-multipleApply constant multiple rule to both terms.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \tan{\left(4 x - 1 \right)}} - \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \tan{\left(4 x - 1 \right)}} - \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]sumDifferentiate the argument of the first logarithm.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \tan{\left(4 x - 1 \right)}} - \frac{5 \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \tan{\left(4 x - 1 \right)}} - \frac{5 \tan{\left(4 x - 1 \right)} \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{5 \sec^{2}{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \tan{\left(4 x - 1 \right)}} - \frac{5 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]chainApply the chain rule to the tangent function.≈ Checked numerically
- \[ = \frac{5 \sec^{2}{\left(4 x - 1 \right)}}{\tan{\left(4 x - 1 \right)}} - \frac{5 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1} \]derivative algebra algebraDifferentiate the inner linear function 4x - 1. Multiply the constants in each term. Simplify the coefficients.✓ Proved
- \[ = 5 \left(\frac{1}{\tan{\left(4 x - 1 \right)}} - \frac{\tan{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1}\right) \sec^{2}{\left(4 x - 1 \right)} \]algebraFactor out the common term.✓ Proved
- \[ = \frac{5 \sec^{2}{\left(4 x - 1 \right)}}{\left(\tan^{2}{\left(4 x - 1 \right)} + 1\right) \tan{\left(4 x - 1 \right)}} \]algebra algebra simplifyFind a common denominator inside the parentheses. Simplify the numerator. Cancel the squared tangent terms.✓ Proved
- \[ = \frac{5}{\left(\tan^{2}{\left(4 x - 1 \right)} + 1\right) \cos^{2}{\left(4 x - 1 \right)} \tan{\left(4 x - 1 \right)}} \]rewrite algebraRewrite secant in terms of cosine. Combine the terms into a single fraction.✓ Proved
Answer \( \frac{5}{\tan{\left(4 x - 1 \right)}} \)
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.
Lines: 16 proved, 2 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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*x - 1)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (5*tan(4*x - 1)**2 - 5*sec(4*x - 1)**2 + 5)/(tan(4*x - 1)**3 + tan(4*x - 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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(4*x - 1) = 0 undefined where tan(4*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 sec has poles at odd multiples of pi/2 undefined where tan(4*x - 1) = 0 undefined where tan(4*x - 1)**2 + 1 = 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(4*x - 1) = 0 undefined where tan(4*x - 1)**2 + 1 = 0 undefined where cos(4*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(4*x - 1) = 0 undefined where tan(4*x - 1)**2 + 1 = 0 undefined where cos(4*x - 1) = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 5*(-(tan(4*x - 1)**2 + 1)*cos(4*x - 1)**2 + 1)/((tan(4*x - 1)**2 + 1)*cos(4*x - 1)**2*tan(4*x - 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(4*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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is valid and leads to the correct result.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is valid and leads to the correct result.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification is valid and leads to the correct final answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 17 combines the terms into a single fraction but fails to cancel the factor θ²+1 with σ² (since σ²=1/σ²). The correct simplification gives 5/tan(4*x-1).qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 2 applies the sum rule to split the derivative, but is labeled 'constant-multiple'. Step 3 pulls out constants and is labeled 'constant-multiple', but Step 2 already performed the splitting. The label in Step 2 is incorrect for the operation performed (splitting a sum).deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification is algebraically correct and matches the stated answer.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 15’s note claims to cancel squared tangent terms, but the expression 5*sec^2*(1/(tan*(tan^2+1))) contains no squared tan terms to cancel. This misleading note could confuse a student.deepseek-r1:70b: pass 2026-09-17
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.