Derivative of \( \displaystyle \frac{5 \ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} \)
Problem 2.873 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\tan^{2}{\left(3 x + 1 \right)} + 1 \right)}}{6} \]constant-multiplePull out the constant coefficient.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)}{6 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(3 x + 1 \right)}\right)}{6 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan^{2}{\left(3 x + 1 \right)}}{6 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]constant algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
- \[ = \frac{5 \tan{\left(3 x + 1 \right)} \frac{d}{d x} \tan{\left(3 x + 1 \right)}}{3 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{5 \tan{\left(3 x + 1 \right)} \sec^{2}{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right)}{3 \left(\tan^{2}{\left(3 x + 1 \right)} + 1\right)} \]trigDifferentiate the tangent function.≈ Checked numerically
- \[ = \frac{5 \tan{\left(3 x + 1 \right)} \sec^{2}{\left(3 x + 1 \right)}}{\tan^{2}{\left(3 x + 1 \right)} + 1} \]chain algebraDifferentiate the inner linear function. Multiply the constants together.✓ Proved
- \[ = \frac{30 \tan{\left(3 x + 1 \right)} \sec^{2}{\left(3 x + 1 \right)}}{6 \tan^{2}{\left(3 x + 1 \right)} + 6} \]algebraDistribute the 5/6 into the expression.✓ Proved
- \[ = \frac{5 \tan{\left(3 x + 1 \right)} \sec^{2}{\left(3 x + 1 \right)}}{\tan^{2}{\left(3 x + 1 \right)} + 1} \]simplifySimplify the fraction by canceling the 6.✓ Proved
Answer \( 5 \tan{\left(3 x + 1 \right)} \)
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: 11 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 undefined where tan(3*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(3*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(3*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(3*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(3*x + 1)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*(tan(3*x + 1)**2 - sec(3*x + 1)**2 + 1)*tan(3*x + 1)/(tan(3*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(3*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(3*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(3*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(3*x + 1)**2 + 1 = 0 undefined where 6*tan(3*x + 1)**2 + 6 = 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 6*tan(3*x + 1)**2 + 6 = 0 undefined where tan(3*x + 1)**2 + 1 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 5*(-tan(3*x + 1)**2 + sec(3*x + 1)**2 - 1)*tan(3*x + 1)/(tan(3*x + 1)**2 + 1); numeric agreement only, at 24 of 24 sampled points 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: fail (error) — The final answer is mathematically incorrect; the derivative simplifies to 5*tan(3*x + 1)/2, not 5*tan(3*x + 1). The solution fails to cancel the factor of 2 arising from the power rule and the trigonometric identity sec^2 = 1 + tan^2.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final answer is mathematically incorrect; the derivative simplifies to 5*tan(3*x + 1)/2, not 5*tan(3*x + 1). The solution fails to cancel the factor of 2 arising from the power rule and the trigonometric identity sec^2 = 1 + tan^2.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution stops at step 12 without simplifying the expression to the stated answer. Since sec^2(u) = tan^2(u) + 1, the term sec(3*x + 1)**2 / (tan(3*x + 1)**2 + 1) simplifies to 1, yielding the final answer 5*tan(3*x + 1). The provided steps do not reach the stated answer.gpt-oss:20b: fail (misleading) 2026-09-26 — The final expression 5*tan(3*x+1) is correct, but the solution stops at 5*tan(3*x+1)*sec(3*x+1)**2/(tan(3*x+1)**2+1) and does not simplify sec^2/(tan^2+1)=1. This omission misleads a student into thinking the derivative is more complicated than it is.
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.