Derivative of \( \displaystyle x^{x} \)
Problem 2.486 · medium Video
Differentiate \( \displaystyle f(x) = x^{x} \).
- \[ \frac{d}{d x} x^{x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{x \ln{\left(x \right)}} \]rewriteRewrite the base-exponent form using the exponential and logarithm.✓ Checked independently
- \[ = e^{x \ln{\left(x \right)}} \frac{d}{d x} x \ln{\left(x \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x\right) e^{x \ln{\left(x \right)}} \]derivativeDifferentiate the terms in the product.✓ Proved
- \[ = \left(\ln{\left(x \right)} + 1\right) e^{x \ln{\left(x \right)}} \]derivative algebraCompute the derivatives of x and log(x). Simplify the expression inside the parentheses.✓ Proved
- \[ = x^{x} \left(\ln{\left(x \right)} + 1\right) \]simplifySubstitute back the original form of the exponential.✓ Checked independently
Answer \( x^{x} \left(\log{\left(x \right)} + 1\right) \)
Lines: 6 proved, 2 checked by independent solution. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left (x**x - exp(x*log(x)))*(log(x) + 1); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: x**x rewritten as exp(x*log(x)) (Pow.rewrite(exp)=exp(x*log(x))); derivatives of both sides agree: False ( checks/2.486/line2-5db57c53.py)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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left (-x**x + exp(x*log(x)))*(log(x) + 1); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: exp(x*log(x)) <-> x**x via rewrite: True True True ( checks/2.486/line7-5836b687.py)log is undefined for non-positive arguments |
| 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 (error) — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(x*log(x)) should be exp(x*log(x)) * Derivative(x*log(x), x), which is a chain rule application, not a product rule.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'product', but the step applies the chain rule to differentiate exp(u). The product rule is not applied to the outer exponential function; it is applied in the subsequent step to the inner term. Labeling the chain rule application as 'product' is a factual error regarding the rule used.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct throughout and SymPy proved the final answer. Line 3 is labeled 'product' but applies the chain rule to exp(x*log(x)). The product rule is only applied in line 4, so a student would learn the wrong rule name.
gpt-oss:20b: uphold — Line 3 only pulls out exp(u) times Derivative(u) without expanding the inner derivative, so the rule applied is chain (or exponential), not product.qwen3.6:27b-mlx: uphold — The product rule is first used in line 4; line 3 is the outer exp(u) differentiation, so the 'product' label is a wrong rule name.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The math is correct throughout and SymPy proved the final answer. Line 3 is labeled 'product' but applies the chain rule to exp(x*log(x)). The product rule is only applied in line 4, so a student would learn the wrong rule name.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'product', but the step applies the chain rule to differentiate exp(u). The product rule is not applied to the outer exponential function; it is applied in the subsequent step to the inner term. Labeling the chain rule application as 'product' is a factual error regarding the rule used.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(x*log(x)) should be exp(x*log(x)) * Derivative(x*log(x), x), which is a chain rule application, not a product rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — 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 term x*log(x). The labels are swapped.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 applies the chain rule (derivative of exp(g(x))) but is labeled as "product", which is incorrect labeling of the rule used.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'product' but applies the chain rule to differentiate the outer exponential function. Step 4 is labeled 'derivative' but applies the product rule to the inner term x*log(x). The labels are swapped.gpt-oss:20b: fail (style) 2026-09-20 — 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 terms, so the rule name is wrong. This mislabeling could mislead a student about which rule to apply.
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.