Home›Calculus 1›Logarithmic differentiation›Problem 2.939

Derivative of \( \displaystyle x^{\sqrt{x}} \)

Problem 2.939 · hard Video

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 \( \frac{x^{\sqrt{x} - \frac{1}{2}} \left(\ln{\left(x \right)} + 2\right)}{2} \)

Top: the graph of f = x^(sqrt(x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x^(sqrt(x))·(log(x)/(2sqrt(x)) + 1/sqrt(x)).

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. 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 numerically	sympy 1.14.0	sympy 1.14.0: simplify left (x*(sqrt(x)) - exp(sqrt(x)log(x)))(log(x) + 2)/(2sqrt(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)/(2sqrt(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 (error) — Step 3 applies the chain rule, not the product rule. The label and note incorrectly state a product rule was used on the exponent. This mislabels the applied rule and misleads the reader.
qwen3.6:27b-mlx: fail (error) — Step 1 is labeled 'rewrite' but performs the differentiation operation (Derivative(..., x)), which violates the contract that 'derivative' is the label for unfolding d/dx. Step 2 is labeled 'derivative' but performs a substitution/rewrite of the base function, not the differentiation of the exponential form. The labels are swapped and incorrect.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 1 is labeled 'rewrite' but performs the differentiation operation (Derivative(..., x)), which violates the contract that 'derivative' is the label for unfolding d/dx. Step 2 is labeled 'derivative' but performs a substitution/rewrite of the base function, not the differentiation of the exponential form. The labels are swapped and incorrect.
gpt-oss:20b: fail (error) 2026-09-28 — Step 3 applies the chain rule, not the product rule. The label and note incorrectly state a product rule was used on the exponent. This mislabels the applied rule and misleads the reader.
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 2026-09-27
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, 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 2026-09-28 with SymPy 1.14.0.