Derivative of \( \displaystyle \frac{3 \ln{\left(\tan^{2}{\left(4 x + 1 \right)} + 1 \right)}}{8} \)
Problem 2.1075 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan^{2}{\left(4 x + 1 \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \frac{3 \ln{\left(\tan^{2}{\left(4 x + 1 \right)} + 1 \right)}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x + 1 \right)} + 1 \right)}}{8} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\tan^{2}{\left(4 x + 1 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x + 1 \right)} + 1\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan^{2}{\left(4 x + 1 \right)}}{8 \left(\tan^{2}{\left(4 x + 1 \right)} + 1\right)} \]constantThe derivative of the constant 1 is 0.✓ Proved
- \[ = \frac{3 \tan{\left(4 x + 1 \right)} \frac{d}{d x} \tan{\left(4 x + 1 \right)}}{4 \left(\tan^{2}{\left(4 x + 1 \right)} + 1\right)} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{3 \tan{\left(4 x + 1 \right)} \sec^{2}{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{4 \left(\tan^{2}{\left(4 x + 1 \right)} + 1\right)} \]chainApply the chain rule to the tangent function.≈ Checked numerically
- \[ = \frac{3 \tan{\left(4 x + 1 \right)} \sec^{2}{\left(4 x + 1 \right)}}{\tan^{2}{\left(4 x + 1 \right)} + 1} \]derivative algebra simplifyDifferentiate the innermost linear function. Multiply the constants together. Simplify the constant coefficients and combine terms.✓ Proved
Answer \( 3 \tan{\left(4 x + 1 \right)} \)
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.
Lines: 8 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(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 | ≈ 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)/(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 |
| 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)**2 + 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)**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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 3*(-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 |
| 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) — Step 2 is labeled 'logarithmic' but does not apply the logarithmic derivative rule; it merely rewrites the expression, which should be labeled 'rewrite' or 'algebra'. Step 3 is labeled 'sum' but actually applies the chain rule to the logarithm, which is a mislabeling of the differentiation rule applied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 2 is labeled 'logarithmic' but does not apply the logarithmic derivative rule; it merely rewrites the expression, which should be labeled 'rewrite' or 'algebra'. Step 3 is labeled 'sum' but actually applies the chain rule to the logarithm, which is a mislabeling of the differentiation rule applied.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 3 is labeled 'sum' but actually applies the chain rule to the logarithm. Step 5 is labeled 'power' but applies the chain rule to the square function (derivative of u^2 is 2u*u'). These labels misidentify the differentiation rules used.gpt-oss:20b: fail (misleading) 2026-09-28 — The final step claims to have simplified the expression, yet it still contains the factor sec(4*x+1)**2/(tan(4*x+1)**2+1). Using the identity tan^2+1=sec^2, this fraction reduces to 1, yielding the correct derivative 3*tan(4*x+1). The solution therefore does not fully simplify the result, misleading a student into thinking the unsimplified form is the final answer.
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.