Derivative of \( \displaystyle \left(x + 1\right)^{x + 1} \)
Problem 2.1280 · medium
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{x + 1} \).
- \[ \frac{d}{d x} \left(x + 1\right)^{x + 1} \]derivative✓ Proved
- \[ = \frac{d}{d x} e^{\left(x + 1\right) \ln{\left(x + 1 \right)}} \]rewriteRewrite the base and exponent 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 term.✓ 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 factor 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 factor 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 form back to the original power form.≈ Checked numerically
Answer \( \left(x + 1\right)^{x + 1} \left(\ln{\left(x + 1 \right)} + 1\right) \)
✓ 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 — The solution correctly applies the one-change-per-step constraint and uses valid labels from the fixed vocabulary. The logic is sound and the notes are accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the one-change-per-step constraint and uses valid labels from the fixed vocabulary. The logic is sound and the notes are accurate.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies the rules in the specified vocabulary. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed (rewrite, chain, product, derivative, algebra, simplify).gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.