Home›Calculus 1›Logarithmic differentiation›Problem 2.941

Derivative of \( \displaystyle x^{x^{2}} \)

Problem 2.941 · hard Video

Differentiate \( \displaystyle f(x) = x^{x^{2}} \).

\[ \frac{d}{d x} x^{x^{2}} \]
rewriteRewrite the base and exponent using the exponential identity.✓ Proved
\[ = \frac{d}{d x} e^{x^{2} \ln{\left(x \right)}} \]
chainApply the chain rule to the exponential function.≈ Checked numerically
\[ = e^{x^{2} \ln{\left(x \right)}} \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
productApply the product rule to the exponent.✓ Proved
\[ = \left(x^{2} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2}\right) e^{x^{2} \ln{\left(x \right)}} \]
derivativeDifferentiate the two parts of the product.✓ Proved
\[ = \left(2 x \ln{\left(x \right)} + x\right) e^{x^{2} \ln{\left(x \right)}} \]
derivative simplifyCompute the derivatives of x**2 and log(x). Simplify the expression inside the parentheses.✓ Proved
\[ = x^{x^{2}} \left(2 x \ln{\left(x \right)} + x\right) \]
simplifySubstitute back the original form of the exponential term.≈ Checked numerically

Answer \( x^{x^{2}} \left(2 x \ln{\left(x \right)} + x\right) \)

Top: the graph of f = x^(x²). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x^(x²)·(2x·log(x) + 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(x(x2) - exp(x2log(x)))(2log(x) + 1); 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
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
7	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left x(-x(x2) + exp(x2log(x)))(2log(x) + 1); numeric agreement only, at 24 of 24 sampled points 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 chain rule as a product rule, and step 4 incorrectly labels the product rule as a derivative. Each step must apply only one rule and use the correct label from the allowed vocabulary.
qwen3.6:27b-mlx: fail (style) — Step 1 applies the exponential identity rewrite but is labeled 'rewrite' while the note describes the action; however, the primary defect is in Step 2. Step 2 claims to apply the 'chain' rule to transform `Derivative(exp(...), x)` into `exp(...) * Derivative(..., x)`, but the label 'chain' is reserved for the differentiation rule itself, whereas the transition from Step 1 to Step 2 is actually the application of the chain rule differentiation. Wait, looking closer: Step 1 is `Derivative(x**(x**2), x)`. Step 2 is `Derivative(exp(x**2 * log(x)), x)`. The transition from 1 to 2 is a rewrite of the function inside the derivative operator. The label 'rewrite' is correct for Step 1. Step 2 is `Derivative(exp(u), x)`. Step 3 is `exp(u) * Derivative(u, x)`. The transition from 2 to 3 applies the chain rule. The label in Step 3 is 'chain'. This is correct. Let's re-read the contract. 'Each step names its rule'. Step 1 label is 'rewrite'. Step 2 label is 'chain'. Step 2 is `Derivative(exp(...), x)`. Step 3 is `exp(...) * Derivative(..., x)`. The step FROM 2 TO 3 is the application of the chain rule. The label should be on the step that performs the change. Usually, the label describes the transition. If Step 2 is the state after Step 1, then the label on Step 2 describes the transition 1->2. Transition 1->2 is a rewrite. Label is 'rewrite'. Correct. Transition 2->3 is chain rule. Label on Step 3 is 'chain'. Correct. Transition 3->4 is product rule. Label on Step 4 is 'product'. Correct. Transition 4->5 is derivative of x^2 and log(x). Label on Step 5 is 'derivative'. Correct. Transition 5->6 is simplify x^2 * 1/x to x. Label on Step 6 is 'simplify'. Correct. Transition 6->7 is rewrite exp back to power. Label on Step 7 is 'simplify'. This is a rewrite, not a simplification. 'Simplify' usually implies algebraic reduction. Converting exp(log) back to power is a 'rewrite'. However, 'simplify' is often used broadly. Is it a defect? The contract says 'simplify' is a presentational step. Rewriting exp(u) as e^u is not simplifying, it's changing form. But wait, Step 7 goes from `exp(x**2 * log(x))` to `x**(x**2)`. This is the inverse of the rewrite in Step 1. It is a rewrite. Labeling it 'simplify' is slightly inaccurate but 'rewrite' is not in the list of 'bad' labels, it's in the vocabulary. The vocabulary includes 'rewrite'. So using 'simplify' for a rewrite is a style defect. Also, Step 1 note says 'Rewrite the base...'. Step 1 label is 'rewrite'. This is fine. Step 7 label is 'simplify'. This is a style defect because it's a structural rewrite, not an algebraic simplification. Let's look for a harder error. Step 4: `Derivative(x**2, x) * log(x) + x**2 * Derivative(log(x), x)`. This is the result of applying product rule to `x**2 * log(x)`. The label on Step 4 is 'product'. This is correct. Step 5: `2*x * log(x) + x**2 * (1/x)`. This computes the derivatives. Label 'derivative'. Correct. Step 6: `2*x * log(x) + x`. Simplifies `x**2 * 1/x`. Label 'simplify'. Correct. Step 7: `x**(x**2) * (...)`. Rewrites `exp(x**2 * log(x))` back to `x**(x**2)`. Label 'simplify'. This is a rewrite. The label 'rewrite' exists in the vocabulary. Using 'simplify' is a style defect. Is there a multi-rule step? Step 4 applies product rule AND derivative? No, Step 4 shows the structure of the product rule application, leaving the inner derivatives as `Derivative(...)`. Step 5 computes those derivatives. This is granular. Step 2 to 3: Chain rule. Step 3 to 4: Product rule. Step 4 to 5: Derivative of components. Step 5 to 6: Simplify. Step 6 to 7: Rewrite. The label 'simplify' in Step 7 is the only potential issue. It's a style defect. I will mark it as fail/style.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 1 applies the exponential identity rewrite but is labeled 'rewrite' while the note describes the action; however, the primary defect is in Step 2. Step 2 claims to apply the 'chain' rule to transform `Derivative(exp(...), x)` into `exp(...) * Derivative(..., x)`, but the label 'chain' is reserved for the differentiation rule itself, whereas the transition from Step 1 to Step 2 is actually the application of the chain rule differentiation. Wait, looking closer: Step 1 is `Derivative(x**(x**2), x)`. Step 2 is `Derivative(exp(x**2 * log(x)), x)`. The transition from 1 to 2 is a rewrite of the function inside the derivative operator. The label 'rewrite' is correct for Step 1. Step 2 is `Derivative(exp(u), x)`. Step 3 is `exp(u) * Derivative(u, x)`. The transition from 2 to 3 applies the chain rule. The label in Step 3 is 'chain'. This is correct. Let's re-read the contract. 'Each step names its rule'. Step 1 label is 'rewrite'. Step 2 label is 'chain'. Step 2 is `Derivative(exp(...), x)`. Step 3 is `exp(...) * Derivative(..., x)`. The step FROM 2 TO 3 is the application of the chain rule. The label should be on the step that performs the change. Usually, the label describes the transition. If Step 2 is the state after Step 1, then the label on Step 2 describes the transition 1->2. Transition 1->2 is a rewrite. Label is 'rewrite'. Correct. Transition 2->3 is chain rule. Label on Step 3 is 'chain'. Correct. Transition 3->4 is product rule. Label on Step 4 is 'product'. Correct. Transition 4->5 is derivative of x^2 and log(x). Label on Step 5 is 'derivative'. Correct. Transition 5->6 is simplify x^2 * 1/x to x. Label on Step 6 is 'simplify'. Correct. Transition 6->7 is rewrite exp back to power. Label on Step 7 is 'simplify'. This is a rewrite, not a simplification. 'Simplify' usually implies algebraic reduction. Converting exp(log) back to power is a 'rewrite'. However, 'simplify' is often used broadly. Is it a defect? The contract says 'simplify' is a presentational step. Rewriting exp(u) as e^u is not simplifying, it's changing form. But wait, Step 7 goes from `exp(x**2 * log(x))` to `x**(x**2)`. This is the inverse of the rewrite in Step 1. It is a rewrite. Labeling it 'simplify' is slightly inaccurate but 'rewrite' is not in the list of 'bad' labels, it's in the vocabulary. The vocabulary includes 'rewrite'. So using 'simplify' for a rewrite is a style defect. Also, Step 1 note says 'Rewrite the base...'. Step 1 label is 'rewrite'. This is fine. Step 7 label is 'simplify'. This is a style defect because it's a structural rewrite, not an algebraic simplification. Let's look for a harder error. Step 4: `Derivative(x**2, x) * log(x) + x**2 * Derivative(log(x), x)`. This is the result of applying product rule to `x**2 * log(x)`. The label on Step 4 is 'product'. This is correct. Step 5: `2*x * log(x) + x**2 * (1/x)`. This computes the derivatives. Label 'derivative'. Correct. Step 6: `2*x * log(x) + x`. Simplifies `x**2 * 1/x`. Label 'simplify'. Correct. Step 7: `x**(x**2) * (...)`. Rewrites `exp(x**2 * log(x))` back to `x**(x**2)`. Label 'simplify'. This is a rewrite. The label 'rewrite' exists in the vocabulary. Using 'simplify' is a style defect. Is there a multi-rule step? Step 4 applies product rule AND derivative? No, Step 4 shows the structure of the product rule application, leaving the inner derivatives as `Derivative(...)`. Step 5 computes those derivatives. This is granular. Step 2 to 3: Chain rule. Step 3 to 4: Product rule. Step 4 to 5: Derivative of components. Step 5 to 6: Simplify. Step 6 to 7: Rewrite. The label 'simplify' in Step 7 is the only potential issue. It's a style defect. I will mark it as fail/style.
gpt-oss:20b: fail (error) 2026-09-28 — Step 3 incorrectly labels the chain rule as a product rule, and step 4 incorrectly labels the product rule as a derivative. Each step must apply only one rule and use the correct label from the allowed vocabulary.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 1 applies the exponential identity rewrite but also attempts to differentiate, violating the 'one change per step' rule. Step 2 is labeled 'chain' but performs no differentiation; it is merely a presentation of the rewritten form, making the label incorrect for the operation shown.
gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the chain rule result as a product rule, and step 4 incorrectly labels a product differentiation as a simple derivative. The labels do not match the operations performed.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 1 is labeled 'rewrite' but performs a differentiation (Derivative(...)), which is not a presentational step. Step 2 is labeled 'chain' but the expression is still wrapped in a Derivative operator; it should be labeled 'derivative' to unfold the outer exponential derivative, or the step is logically incomplete.
gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the application of the chain rule as a product rule; the derivative of exp(u) is exp(u)*u', not a product rule on the exponent. The subsequent steps misuse the product rule terminology, leading to a labeling defect.

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.