Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(4 x - 3 \right)} + 1 \right)}}{8} \)
Problem 2.1209 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(4 x - 3 \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(4 x - 3 \right)} + 1 \right)}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x - 3 \right)} + 1 \right)}}{8} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(4 x - 3 \right)}}{8 \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan^{2}{\left(4 x - 3 \right)}}{8 \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)} \]powerApply the power rule to the tangent squared term.✓ Proved
- \[ = \frac{\tan{\left(4 x - 3 \right)} \frac{d}{d x} \tan{\left(4 x - 3 \right)}}{4 \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)} \]chainApply the chain rule to the tangent function.✓ Proved
- \[ = \frac{\tan{\left(4 x - 3 \right)} \sec^{2}{\left(4 x - 3 \right)} \frac{d}{d x} \left(4 x - 3\right)}{4 \left(\tan^{2}{\left(4 x - 3 \right)} + 1\right)} \]chainApply the chain rule to the inner linear function.≈ Checked numerically
- \[ = \frac{\tan{\left(4 x - 3 \right)} \sec^{2}{\left(4 x - 3 \right)}}{\tan^{2}{\left(4 x - 3 \right)} + 1} \]algebra algebra simplify algebraDifferentiate the linear term 4x - 3. Multiply the constants in the numerator. Simplify the constant coefficients. Combine the terms into a single fraction.✓ Proved
- \[ = \tan{\left(4 x - 3 \right)} \]rewrite simplifyUse the identity sec(u)**2 = 1 + tan(u)**2. Cancel the common factor in the numerator and denominator.≈ Checked numerically
Answer \( \tan{\left(4 x - 3 \right)} \)
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 - 3)**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 - 3)**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 - 3)**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 - 3)**2 + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(4*x - 3)**2 - sec(4*x - 3)**2 + 1)*tan(4*x - 3)/(tan(4*x - 3)**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 - 3)**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 - 3)**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 - 3)**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 - 3)**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 - 3)**2 + 1 = 0 |
| 12 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-tan(4*x - 3)**2 + sec(4*x - 3)**2 - 1)*tan(4*x - 3)/(tan(4*x - 3)**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 - 3)**2 + 1 = 0 |
| 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: fail (style) — Step 6 incorrectly labels the operation as "chain"; the derivative of tan(4*x‑3)**2 is obtained via the power rule, not the chain rule. The step also applies a product of 2*tan(4*x‑3) and the derivative of tan, which is a constant multiplication, not a chain rule application. This mislabeling could mislead a student about the correct rule to use.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-29 — Step 6 incorrectly labels the operation as "chain"; the derivative of tan(4*x‑3)**2 is obtained via the power rule, not the chain rule. The step also applies a product of 2*tan(4*x‑3) and the derivative of tan, which is a constant multiplication, not a chain rule application. This mislabeling could mislead a student about the correct rule to use.qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 8 is labeled 'algebra' but performs the differentiation of the linear term 4x-3; it should be labeled 'derivative'. Step 7 is labeled 'chain' but performs the differentiation of tan(u); it should be labeled 'derivative' (or 'trig') as the chain rule application for the inner function happens in the next step.gpt-oss:20b: fail (style) 2026-09-29 — Step 8 is labeled "algebra" but actually applies the derivative rule to the linear function 4*x-3. The correct label should be "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 2026-09-29 with SymPy 1.14.0.