Derivative of \( \displaystyle x^{\frac{1}{x}} \)
Problem 2.936 · hard
Differentiate \( \displaystyle f(x) = x^{\frac{1}{x}} \).
- \[ \frac{d}{d x} x^{\frac{1}{x}} \]rewriteRewrite the function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{\frac{\ln{\left(x \right)}}{x}} \]derivativeApply the derivative rule for the exponential function.≈ Checked numerically
- \[ = e^{\frac{\ln{\left(x \right)}}{x}} \frac{d}{d x} \frac{\ln{\left(x \right)}}{x} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\ln{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \ln{\left(x \right)}}{x}\right) e^{\frac{\ln{\left(x \right)}}{x}} \]derivativeDifferentiate both parts of the product.✓ Proved
- \[ = \left(- \frac{\ln{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right) e^{\frac{\ln{\left(x \right)}}{x}} \]derivative algebraCompute the derivatives of 1/x and log(x). Simplify the terms inside the parentheses.✓ Proved
- \[ = \frac{\left(1 - \ln{\left(x \right)}\right) e^{\frac{\ln{\left(x \right)}}{x}}}{x^{2}} \]algebraCombine the fractions.✓ Proved
- \[ = \frac{x^{\frac{1}{x}} \left(1 - \ln{\left(x \right)}\right)}{x^{2}} \]simplifySubstitute back the original base.≈ Checked numerically
Answer \( x^{-2 + \frac{1}{x}} \left(1 - \ln{\left(x \right)}\right) \)
Lines: 7 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**(1/x)*(1 - log(x)) + (log(x) - 1)*exp(log(x)/x))/x**2; numeric agreement only, at 24 of 24 sampled points undefined where x = 0 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 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (x**(1/x)*(log(x) - 1) + (1 - log(x))*exp(log(x)/x))/x**2; 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 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 3 incorrectly labels the application of the chain rule for the exponential function as a product rule. The derivative of exp(u) is exp(u)*u', which is a chain rule application, not a product rule. The label should be "chain" (or "derivative" with the appropriate rule).qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'product' but applies the chain rule to the outer exponential function. Step 4 is labeled 'derivative' but applies the product rule to the inner exponent. The labels are swapped.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 is labeled 'product' but applies the chain rule to the outer exponential function. Step 4 is labeled 'derivative' but applies the product rule to the inner exponent. The labels are swapped.gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the application of the chain rule for the exponential function as a product rule. The derivative of exp(u) is exp(u)*u', which is a chain rule application, not a product rule. The label should be "chain" (or "derivative" with the appropriate rule).qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 is labeled 'product' but applies the chain rule to the outer exponential function; the product rule is applied in the subsequent step (Step 4) to the exponent. Additionally, Step 2 is labeled 'derivative' but performs no differentiation, merely rewriting the expression.gpt-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. This mislabeling misleads the student about which rule is being used.
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.