Derivative of \( \displaystyle \ln{\left(x \right)}^{x} \)
Problem 2.477 · hard Video
Differentiate \( \displaystyle f(x) = \ln{\left(x \right)}^{x} \).
- \[ \frac{d}{d x} \ln{\left(x \right)}^{x} \]rewriteRewrite the function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{x \ln{\left(\ln{\left(x \right)} \right)}} \]chainApply the chain rule.✓ Checked independently
- \[ = e^{x \ln{\left(\ln{\left(x \right)} \right)}} \frac{d}{d x} x \ln{\left(\ln{\left(x \right)} \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(\ln{\left(x \right)} \right)} + \ln{\left(\ln{\left(x \right)} \right)} \frac{d}{d x} x\right) e^{x \ln{\left(\ln{\left(x \right)} \right)}} \]productDistribute the derivative across the product.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(\ln{\left(x \right)} \right)} + \ln{\left(\ln{\left(x \right)} \right)}\right) e^{x \ln{\left(\ln{\left(x \right)} \right)}} \]constantThe derivative of x is 1.✓ Proved
- \[ = \left(\frac{x \frac{d}{d x} \ln{\left(x \right)}}{\ln{\left(x \right)}} + \ln{\left(\ln{\left(x \right)} \right)}\right) e^{x \ln{\left(\ln{\left(x \right)} \right)}} \]chainApply the chain rule to log(log(x)).✓ Proved
- \[ = \left(\ln{\left(\ln{\left(x \right)} \right)} + \frac{1}{\ln{\left(x \right)}}\right) e^{x \ln{\left(\ln{\left(x \right)} \right)}} \]derivative algebraThe derivative of log(x) is 1/x. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(\ln{\left(\ln{\left(x \right)} \right)} + \frac{1}{\ln{\left(x \right)}}\right) \ln{\left(x \right)}^{x} \]simplifyConvert the exponential form back to the original base.✓ Checked independently
Answer \( \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}^{x} \)
Lines: 8 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 (log(x)*log(log(x)) + 1)*(-exp(x*log(log(x))) + log(x)**x)/log(x); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: derivatives agree after substituting exp(x*log(log(x))) = log(x)**x; identity log(b**x)=x*log(b) checked: True, diff zero: True ( checks/2.477/line2-28acaa22.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 undefined where log(x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x) = 0 |
| 9 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left (log(x)*log(log(x)) + 1)*(exp(x*log(log(x))) - log(x)**x)/log(x); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: line 9 is line 8 with exp(x*log(log(x))) replaced by log(x)**x (b**x = exp(x*log b)); log identity: True, diff zero: True ( checks/2.477/line9-a4ca1fb7.py)log is undefined for non-positive arguments undefined where log(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 log(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 chain rule as a product rule, and step 4 incorrectly labels the sum of derivatives as a product rule. These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'product' but performs algebraic distribution of the derivative operator, not the product rule for differentiation. Step 5 is labeled 'constant' but computes the derivative of x, which should be labeled 'derivative'.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct throughout and the final answer matches. The labels on lines 2 and 3 are shifted: the step 1→2 is a rewrite (log(x)**x to exp(x*log(log(x)))) but is labeled 'chain', and the step 2→3 is the chain rule but is labeled 'product'. Those are wrong rule names.
gpt-oss:20b: uphold — The step into line 3 (exp(u) becomes exp(u)*u') is the chain rule, not the product rule, so its label is wrong; the complaint about line 4 is mistaken because 3→4 really is the product rule.qwen3.6:27b-mlx: dismiss — Line 4 does apply the product rule to x*log(log(x)), and 'constant' versus 'derivative' for d/dx x = 1 is an acceptable either-label case.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The math is correct throughout and the final answer matches. The labels on lines 2 and 3 are shifted: the step 1→2 is a rewrite (log(x)**x to exp(x*log(log(x)))) but is labeled 'chain', and the step 2→3 is the chain rule but is labeled 'product'. Those are wrong rule names.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'product' but performs algebraic distribution of the derivative operator, not the product rule for differentiation. Step 5 is labeled 'constant' but computes the derivative of x, which should be labeled 'derivative'.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the chain rule as a product rule, and step 4 incorrectly labels the sum of derivatives as a product rule. These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule step, not a product rule. The note also incorrectly states that the product rule is applied to the exponent.qwen3.6:27b-mlx: inconclusive 2026-09-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "style", "notes": "Step 1 is labeled 'rewrite' but performs the differentiation setup; the label should be 'rewrite' for the algebraic transformation, but the step desgpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule application, not a product rule.
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.