Derivative of \( \displaystyle \frac{\sqrt[3]{x}}{\sec{\left(x \right)}} \)
Problem 2.935 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt[3]{x}}{\sec{\left(x \right)}} \).
- \[ \frac{d}{d x} \frac{\sqrt[3]{x}}{\sec{\left(x \right)}} \]derivative rewriteStart with the derivative of the function. Rewrite the reciprocal of sec(x) as cos(x).✓ Proved
- \[ = \frac{d}{d x} \sqrt[3]{x} \cos{\left(x \right)} \]algebraSimplify the expression.✓ Proved
- \[ = \sqrt[3]{x} \frac{d}{d x} \cos{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} \sqrt[3]{x} \]productApply the product rule.✓ Proved
- \[ = \sqrt[3]{x} \frac{d}{d x} \cos{\left(x \right)} + \frac{\cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]powerDifferentiate the first term using the power rule.✓ Proved
- \[ = - \sqrt[3]{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]trig algebraDifferentiate the second term. Simplify the expression.✓ Proved
- \[ = \frac{- 3 x \sin{\left(x \right)} + \cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]simplifyCombine into a single fraction.✓ Proved
Answer \( \frac{- x \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{3}}{x^{\frac{2}{3}}} \)
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. How we verify
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 sec has poles at odd multiples of pi/2 undefined where sec(x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where sec(x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 8 incorrectly combines the terms into a single fraction. The second term should be expressed as \n\((x^{1/3}\sin x)\) over \(3x^{2/3}\), which yieldsqwen3.6:27b-mlx: pass — The solution correctly applies the product rule, power rule, and trigonometric derivative rules. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the product rule, power rule, and trigonometric derivative rules. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 8 incorrectly combines the terms into a single fraction. The second term should be expressed as \n\((x^{1/3}\sin x)\) over \(3x^{2/3}\), which yieldsqwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
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.