Derivative of \( \displaystyle \frac{\tan{\left(4 x \right)}}{4} \)
Problem 2.345 · easy
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(4 x \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\tan{\left(4 x \right)}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(4 x \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sec^{2}{\left(4 x \right)} \frac{d}{d x} 4 x}{4} \]chainApply the chain rule to the tangent function.✓ Proved
- \[ = \sec^{2}{\left(4 x \right)} \]derivative algebraDifferentiate the inner function 4*x. Simplify the expression by multiplying the constants.✓ Proved
- \[ = \frac{1}{\cos^{2}{\left(4 x \right)}} \]rewriteRewrite sec(4*x) in terms of cos(4*x).✓ Proved
- \[ = \sec^{2}{\left(4 x \right)} \]simplifyThe expression is already simplified.✓ Proved
Answer \( \tan^{2}{\left(4 x \right)} + 1 \)
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.
Every line of this solution was proved by the computer algebra system SymPy. 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 tan has poles at odd multiples of pi/2 |
| 3 | ✓ 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where cos(4*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x) = 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 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer `tan(4*x)**2 + 1` is mathematically incorrect; the derivative of `tan(4*x)/4` is `sec(4*x)**2`, not `tan(4*x)**2 + 1` (which would
Every verdict on record (13)
qwen3.6:27b-mlx: inconclusive 2026-09-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer `tan(4*x)**2 + 1` is mathematically incorrect; the derivative of `tan(4*x)/4` is `sec(4*x)**2`, not `tan(4*x)**2 + 1` (which wouldgpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — The stated answer is tan(4*x)**2 + 1, but the derivation correctly concludes with sec(4*x)**2. Although these are equal via the identity sec^2 = 1 + tan^2, the final step of the derivation (step 7) is sec(4*x)**2, not tan(4*x)**2 + 1. The solution fails to perform the algebraic rewrite to match the stated answer, making the stated answer inconsistent with the final derived line.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — The stated answer is tan(4*x)**2 + 1, but the final step of the derivation is sec(4*x)**2. Although these are mathematically equivalent, the solution fails to perform the final rewrite step to match the stated answer, leaving the derived result inconsistent with the claimed output.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — The stated answer is tan(4*x)**2 + 1, but the steps conclude with sec(4*x)**2. Although these are equivalent, the solution fails to perform the final rewrite step to match the stated answer, leaving the derivation incomplete relative to the claimed result.deepseek-r1:70b: fail (style) 2026-09-19 — Step 7 is redundant and doesn't contribute to simplification. The solution doesn't explicitly show the equivalence between sec^2 and tan^2 +1, which could be misleading.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-18 — The stated answer is tan(4*x)**2 + 1, but the derivation concludes with sec(4*x)**2. While these are trigonometrically equivalent, the solution fails to perform the final rewrite step to match the stated answer, leaving the last line inconsistent with the claimed result.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18
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.