Derivative of \( \displaystyle \sqrt[3]{x} + x^{4} \)
Problem 2.636 · medium
Differentiate \( \displaystyle f(x) = \sqrt[3]{x} + x^{4} \).
- \[ \frac{d}{d x} \left(\sqrt[3]{x} + x^{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \sqrt[3]{x} + \frac{d}{d x} x^{4} \]sum rewriteApply the sum rule. Rewrite the first term using the exponential and logarithm form.✓ Proved
- \[ = \frac{d}{d x} x^{4} + \frac{1}{3 x^{\frac{2}{3}}} \]chain algebraDifferentiate the rewritten term using the chain rule. Simplify the exponent.✓ Proved
- \[ = 4 x^{3} + \frac{1}{3 x^{\frac{2}{3}}} \]power simplifyDifferentiate the second term using the power rule. Simplify the final expression.✓ Proved
Answer \( 4 x^{3} + \frac{1}{3 x^{\frac{2}{3}}} \)
✓ 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 undefined where x = 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: passqwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'chain', but the step applies the power rule to the rewritten exponential form. While the power rule can be derived via the chain rule, the label 'power' is the standard and more direct description for differentiating x^n. Furthermore, Step 3 rewrites x^(1/3) as exp((1/3)log(x)), and Step 4 differentiates it to (1/3)x^(-2/3) in one go, which technically combines the chain rule application with the simplification of the resulting expression, violating the 'one thing per step' principle if interpreted strictly, but the primary issue is the misleading label 'chain' for a standard power rule application.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'chain', but the step applies the power rule to the rewritten exponential form. While the power rule can be derived via the chain rule, the label 'power' is the standard and more direct description for differentiating x^n. Furthermore, Step 3 rewrites x^(1/3) as exp((1/3)log(x)), and Step 4 differentiates it to (1/3)x^(-2/3) in one go, which technically combines the chain rule application with the simplification of the resulting expression, violating the 'one thing per step' principle if interpreted strictly, but the primary issue is the misleading label 'chain' for a standard power rule application.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum rule, rewrites the fractional power for differentiation, and applies the power rule to both terms. The labels accurately reflect the operations performed, and the final simplification is correct.gpt-oss:20b: pass 2026-09-21
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.