Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \)
Problem 2.944 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{\tan{\left(4 x - 1 \right)} \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{\tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]trigApply the derivative rule for tangent.≈ Checked numerically
- \[ = \frac{\tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1} \]derivative algebra algebraDifferentiate the inner linear function. Multiply the constants together. Simplify the expression by canceling the 8 and 1/8.✓ Proved
- \[ = \tan{\left(4 x - 1 \right)} \]algebra algebra simplifyUse the trigonometric identity sec(u)^2 = 1 + tan(u)^2. Distribute the tangent term in the numerator. Cancel the common term (tan(4*x - 1)**2 + 1) from the numerator and denominator.≈ Checked numerically
Answer \( \tan{\left(4 x - 1 \right)} \)
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.
✓ Nihil obstat Lines: 12 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 undefined where tan(4*x - 1)**2 + 1 = 0 |
| 4 | ✓ 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)**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)**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)**2 + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(4*x - 1)**2 - sec(4*x - 1)**2 + 1)*tan(4*x - 1)/(tan(4*x - 1)**2 + 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)**2 + 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(4*x - 1)**2 + 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(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)**2 + 1 = 0 |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-tan(4*x - 1)**2 + sec(4*x - 1)**2 - 1)*tan(4*x - 1)/(tan(4*x - 1)**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(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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 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 |
| 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: pass — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary. The final simplification is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary. The final simplification is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final simplification is mathematically sound.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.