Derivative of \( \displaystyle - \frac{\ln{\left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)} \right)}}{4} \)
Problem 2.1135 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)} \right)}}{4}\right) \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)} \right)}}{4} \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}\right)}{4 \left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = - \frac{\frac{d}{d x} \tan{\left(4 x - 3 \right)} + \frac{d}{d x} \sec{\left(4 x - 3 \right)}}{4 \left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}\right)} \]chainApply the chain rule to each term.✓ Proved
- \[ = - \frac{4 \tan{\left(4 x - 3 \right)} \sec{\left(4 x - 3 \right)} + 4 \sec^{2}{\left(4 x - 3 \right)}}{4 \left(\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}\right)} \]constantDifferentiate the trigonometric functions.≈ Checked numerically
- \[ = - \frac{\tan{\left(4 x - 3 \right)} \sec{\left(4 x - 3 \right)} + \sec^{2}{\left(4 x - 3 \right)}}{\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}} \]algebraFactor out the 4.✓ Proved
- \[ = \frac{- \tan{\left(4 x - 3 \right)} \sec{\left(4 x - 3 \right)} - \sec^{2}{\left(4 x - 3 \right)}}{\tan{\left(4 x - 3 \right)} + \sec{\left(4 x - 3 \right)}} \]algebraSimplify the product of the fractions.✓ Proved
- \[ = - \sec{\left(4 x - 3 \right)} \]algebra simplifyFactor out sec(4*x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( - \frac{1}{\cos{\left(4 x - 3 \right)}} \)
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.
Lines: 9 proved, 1 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 sec 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 sec has poles at odd multiples of pi/2 undefined where tan(4*x - 3) + sec(4*x - 3) = 0 |
| 4 | ✓ 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) + sec(4*x - 3) = 0 |
| 5 | ≈ 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) + sec(4*x - 3)); 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) + sec(4*x - 3) = 0 |
| 6 | ✓ 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) + sec(4*x - 3) = 0 |
| 7 | ✓ 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) + sec(4*x - 3) = 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(4*x - 3) + sec(4*x - 3) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec 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 undefined where cos(4*x - 3) = 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) — Step 5 applies two rules at once (derivative of the trigonometric functions and pulling out the constant factor 4). It is incorrectly labeled as "constant"; the correct label should be "derivative" (or a combination of "derivative" and "constant-multiple" in separate steps).qwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'constant' but performs the differentiation of trigonometric functions (derivative of tan and sec) and applies the chain rule for the inner linear term. The label 'constant' is incorrect for this operation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 5 is labeled 'constant' but performs the differentiation of trigonometric functions (derivative of tan and sec) and applies the chain rule for the inner linear term. The label 'constant' is incorrect for this operation.gpt-oss:20b: fail (error) 2026-09-28 — Step 5 applies two rules at once (derivative of the trigonometric functions and pulling out the constant factor 4). It is incorrectly labeled as "constant"; the correct label should be "derivative" (or a combination of "derivative" and "constant-multiple" in separate steps).qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (style) 2026-09-28 — Step 5 incorrectly labels the differentiation of the trigonometric functions as "constant"; the correct rule name is "derivative" (or "trig" if available).
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 2026-09-28 with SymPy 1.14.0.