Derivative of \( \displaystyle \left(x - 1\right)^{x - 1} \)
Problem 2.635 · medium
Differentiate \( \displaystyle f(x) = \left(x - 1\right)^{x - 1} \).
- \[ \frac{d}{d x} \left(x - 1\right)^{x - 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(x - 1\right) \ln{\left(x - 1 \right)}} \]rewriteRewrite the power using the exponential and logarithm.≈ Checked numerically
- \[ = e^{\left(x - 1\right) \ln{\left(x - 1 \right)}} \frac{d}{d x} \left(x - 1\right) \ln{\left(x - 1 \right)} \]chainApply the chain rule.✓ Proved
- \[ = \left(\left(x - 1\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)\right) e^{\left(x - 1\right) \ln{\left(x - 1 \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(\left(x - 1\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)}\right) e^{\left(x - 1\right) \ln{\left(x - 1 \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\ln{\left(x - 1 \right)} + 1\right) e^{\left(x - 1\right) \ln{\left(x - 1 \right)}} \]derivative algebraDifferentiate the second part of the product. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(x - 1\right)^{x - 1} \left(\ln{\left(x - 1 \right)} + 1\right) \]simplifyConvert the exponential back to the original power form.≈ Checked numerically
Answer \( \left(x - 1\right)^{x - 1} \left(\log{\left(x - 1 \right)} + 1\right) \)
✓ Nihil obstat Lines: 7 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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) - exp((x - 1)*log(x - 1)))*(log(x - 1) + 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-(x - 1)**(x - 1) + exp((x - 1)*log(x - 1)))*(log(x - 1) + 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.