Derivative of \( \displaystyle \frac{3 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \)
Problem 2.580 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \).
- \[ \frac{d}{d x} \frac{3 \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)} \right)}}{4} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]sumDifferentiate the sum inside the logarithm.✓ Proved
- \[ = \frac{3 \left(\frac{d}{d x} \tan{\left(4 x + 1 \right)} + \frac{d}{d x} \sec{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{3 \left(4 \tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + 4 \sec^{2}{\left(4 x + 1 \right)}\right)}{4 \left(\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}\right)} \]trigCompute the derivatives of tan(u) and sec(u).≈ Checked numerically
- \[ = \frac{3 \left(\tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + \sec^{2}{\left(4 x + 1 \right)}\right)}{\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}} \]algebraFactor out the common constant 4.✓ Proved
- \[ = \frac{3 \tan{\left(4 x + 1 \right)} \sec{\left(4 x + 1 \right)} + 3 \sec^{2}{\left(4 x + 1 \right)}}{\tan{\left(4 x + 1 \right)} + \sec{\left(4 x + 1 \right)}} \]algebraSimplify the constant factors.✓ Proved
- \[ = 3 \sec{\left(4 x + 1 \right)} \]algebra simplifyFactor the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\cos{\left(4 x + 1 \right)}} \)
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: 9 proved, 1 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 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 + 1) + sec(4*x + 1) = 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 + 1) + sec(4*x + 1) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*(tan(4*x + 1)**2 - sec(4*x + 1)**2 + 1)/(tan(4*x + 1) + sec(4*x + 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) + sec(4*x + 1) = 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 + 1) + sec(4*x + 1) = 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 + 1) + sec(4*x + 1) = 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 + 1) + sec(4*x + 1) = 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 + 1) = 0 |
| 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: fail (style) — Step 3 is labeled 'sum' but applies the logarithmic derivative rule; the label should be 'logarithmic' or 'chain'. Step 4 is labeled 'chain' but applies the sum rule to split the derivative; the label should be 'sum'. The labels for steps 3 and 4 are swapped.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 is labeled 'sum' but applies the logarithmic derivative rule; the label should be 'logarithmic' or 'chain'. Step 4 is labeled 'chain' but applies the sum rule to split the derivative; the label should be 'sum'. The labels for steps 3 and 4 are swapped.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 is labeled 'sum' but applies the logarithmic derivative rule (chain rule for log); the label should be 'logarithmic' or 'chain'. Step 4 is labeled 'chain' but applies the sum rule to split the derivative of a sum; the label should be 'sum'.gpt-oss:20b: pass 2026-09-21
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.