Derivative of \( \displaystyle x^{\sqrt{x}} \)
Problem 2.939 · hard
Differentiate \( \displaystyle f(x) = x^{\sqrt{x}} \).
- \[ \frac{d}{d x} x^{\sqrt{x}} \]rewriteRewrite the function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{\sqrt{x} \ln{\left(x \right)}} \]derivativeApply the derivative rule for the exponential function.≈ Checked numerically
- \[ = e^{\sqrt{x} \ln{\left(x \right)}} \frac{d}{d x} \sqrt{x} \ln{\left(x \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\sqrt{x} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} \sqrt{x}\right) e^{\sqrt{x} \ln{\left(x \right)}} \]productApply the product rule to the terms in the parentheses.✓ Proved
- \[ = \left(\frac{\ln{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right) e^{\sqrt{x} \ln{\left(x \right)}} \]derivative algebraDifferentiate the square root and the logarithm. Simplify the expression inside the parentheses.✓ Proved
- \[ = \frac{x^{\sqrt{x}} \left(\ln{\left(x \right)} + 2\right)}{2 \sqrt{x}} \]simplifySubstitute back the original function and simplify.≈ Checked numerically
Answer \( \frac{x^{\sqrt{x} - \frac{1}{2}} \left(\ln{\left(x \right)} + 2\right)}{2} \)
Lines: 6 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (x**(sqrt(x)) - exp(sqrt(x)*log(x)))*(log(x) + 2)/(2*sqrt(x)); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-x**(sqrt(x)) + exp(sqrt(x)*log(x)))*(log(x) + 2)/(2*sqrt(x)); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 3 incorrectly labels the application of the chain rule as "product" and describes it as applying the product rule to the exponent. The correct label should be "chain" (or "exponential" for the outer derivative).qwen3.6:27b-mlx: fail (error) — Step 1 is labeled 'rewrite' but performs a differentiation operation (Derivative(..., x)), which is not a presentational step. Step 2 is labeled 'derivative' but applies the chain rule to the exponential function; 'derivative' is reserved for basic forms, and 'chain' is the correct label for differentiating a composite function.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 1 is labeled 'rewrite' but performs a differentiation operation (Derivative(..., x)), which is not a presentational step. Step 2 is labeled 'derivative' but applies the chain rule to the exponential function; 'derivative' is reserved for basic forms, and 'chain' is the correct label for differentiating a composite function.gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the application of the chain rule as "product" and describes it as applying the product rule to the exponent. The correct label should be "chain" (or "exponential" for the outer derivative).qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 3 incorrectly labels the application of the chain rule for the exponential as a product rule. The derivative of exp(u) is exp(u)·u', not a product of two separate functions, so the rule name is wrong.
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 2026-09-27 with SymPy 1.14.0.