Derivative of \( \displaystyle \left(x + 1\right)^{x} \)
Problem 2.819 · medium
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{x} \).
- \[ \frac{d}{d x} \left(x + 1\right)^{x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{x \ln{\left(x + 1 \right)}} \]rewriteRewrite the function using the exponential and logarithm base change.≈ Checked numerically
- \[ = e^{x \ln{\left(x + 1 \right)}} \frac{d}{d x} x \ln{\left(x + 1 \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)} \frac{d}{d x} x\right) e^{x \ln{\left(x + 1 \right)}} \]productApply the product rule to the exponent expression.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(x + 1 \right)} + \ln{\left(x + 1 \right)}\right) e^{x \ln{\left(x + 1 \right)}} \]constant-multipleSimplify the derivative of x.✓ Proved
- \[ = \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) e^{x \ln{\left(x + 1 \right)}} \]logarithmic algebraDifferentiate the natural logarithm. Simplify the expression.✓ Proved
- \[ = \left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) \]simplifyConvert back to the original base form.≈ Checked numerically
Answer \( \left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \ln{\left(x + 1 \right)}\right) \)
✓ 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 + (x + 1)*log(x + 1))*((x + 1)**x - exp(x*log(x + 1)))/(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 undefined where x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (x + (x + 1)*log(x + 1))*(-(x + 1)**x + exp(x*log(x + 1)))/(x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x + 1 = 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 x + 1 = 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 5 applies the derivative of x to 1, which is a derivative operation, but it is labeled as "constant-multiple". The label should be "derivative"; otherwise the labeling is incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (style) 2026-09-26 — Step 5 applies the derivative of x to 1, which is a derivative operation, but it is labeled as "constant-multiple". The label should be "derivative"; otherwise the labeling is incorrect.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
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.