Derivative of \( \displaystyle \sqrt[3]{2} \sqrt[3]{x} + 16 x^{4} \)
Problem 2.627 · medium
Differentiate \( \displaystyle f(x) = \sqrt[3]{2} \sqrt[3]{x} + 16 x^{4} \).
- \[ \frac{d}{d x} \left(\sqrt[3]{2} \sqrt[3]{x} + 16 x^{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 16 x^{4} + \frac{d}{d x} \sqrt[3]{2} \sqrt[3]{x} \]sumApply the sum rule.✓ Proved
- \[ = \sqrt[3]{2} \frac{d}{d x} \sqrt[3]{x} + \frac{d}{d x} 16 x^{4} \]constant-multipleApply the constant multiple rule to the first term.✓ Proved
- \[ = \sqrt[3]{2} \frac{d}{d x} \sqrt[3]{x} + 16 \frac{d}{d x} x^{4} \]constant-multipleApply the constant multiple rule to the second term.✓ Proved
- \[ = 64 x^{3} + \frac{\sqrt[3]{2}}{3 x^{\frac{2}{3}}} \]power algebra simplifyApply the power rule to both terms. Simplify the coefficients and exponents. Final simplified form.✓ Proved
Answer \( 64 x^{3} + \frac{\sqrt[3]{2}}{3 x^{\frac{2}{3}}} \)
✓ Nihil obstat 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. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 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: fail (style) — Step 5 applies the power rule to both terms in a single line, violating the requirement that each step change only one term. It should be split into two separate steps, one for each term.qwen3.6:27b-mlx: pass — The solution correctly applies the sum rule, constant multiple rule, and power rule in separate steps. The final simplification is algebraically sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum rule, constant multiple rule, and power rule in separate steps. The final simplification is algebraically sound.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 applies the power rule to both terms in a single line, violating the requirement that each step change only one term. It should be split into two separate steps, one for each term.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 5 applies the power rule to both terms in one line, violating the rule that each step must change only one thing. Each application of the power rule should be on a single term, with a separate step for the second term.
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.