Derivative of \( \displaystyle \left(1 + \frac{1}{x}\right)^{x} \)
Problem 2.355 · hard
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{x}\right)^{x} \).
- \[ \frac{d}{d x} \left(1 + \frac{1}{x}\right)^{x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
- \[ = e^{x \ln{\left(1 + \frac{1}{x} \right)}} \frac{d}{d x} x \ln{\left(1 + \frac{1}{x} \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(1 + \frac{1}{x} \right)} + \ln{\left(1 + \frac{1}{x} \right)} \frac{d}{d x} x\right) e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]productApply the product rule to the exponent's derivative.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(1 + \frac{1}{x} \right)} + \ln{\left(1 + \frac{1}{x} \right)}\right) e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]constant-multipleDifferentiate the first factor of the product.✓ Proved
- \[ = \left(\frac{x \frac{d}{d x} \left(1 + \frac{1}{x}\right)}{1 + \frac{1}{x}} + \ln{\left(1 + \frac{1}{x} \right)}\right) e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x} \right)} - \frac{1}{x \left(1 + \frac{1}{x}\right)}\right) e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]derivative algebra algebraDifferentiate the inner term 1/x. Simplify the expression within the parentheses. Simplify the fraction.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x} \right)} - \frac{1}{x + 1}\right) e^{x \ln{\left(1 + \frac{1}{x} \right)}} \]algebraSimplify the denominator inside the parenthesis.✓ Proved
- \[ = \left(1 + \frac{1}{x}\right)^{x} \left(\ln{\left(1 + \frac{1}{x} \right)} - \frac{1}{x + 1}\right) \]simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( \left(1 + \frac{1}{x}\right)^{x} \left(\log{\left(1 + \frac{1}{x} \right)} - \frac{1}{x \left(1 + \frac{1}{x}\right)}\right) \)
Lines: 10 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.
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)**x*((x + 1)*log((x + 1)/x) - 1) + (-(x + 1)*log((x + 1)/x) + 1)*exp(x*log((x + 1)/x)))/(x + 1); numeric agreement only, at 24 of 24 sampled points undefined where x = 0 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 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where 1 + 1/x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 + 1/x = 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 + 1/x = 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 + 1/x = 0 undefined where x = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 1 + 1/x = 0 undefined where x = 0 undefined where x + 1 = 0 |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (((x + 1)/x)**x*(-(x + 1)*log((x + 1)/x) + 1) + ((x + 1)*log((x + 1)/x) - 1)*exp(x*log((x + 1)/x)))/(x + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x + 1 = 0 undefined where 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 x = 0 undefined where 1 + 1/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 5 labels the operation as "constant-multiple" but it actually applies the "derivative" rule to the factor x. The label does not match the rule used.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (10)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 5 labels the operation as "constant-multiple" but it actually applies the "derivative" rule to the factor x. The label does not match the rule used.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' rule application. The label does not name the rule applied.gpt-oss:20b: inconclusive 2026-09-20 — reviewer response could not be parsed: {"verdict":"fail","severity":"style","notes":"Step 5 incorrectly labels the differentiation of \(x\) as a \"constant‑multiple\" rule; it should be a \"derivative\" (or \"sum\" if viewed as a product wqwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 5 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' rule application. The label does not name the rule applied.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 5 is labeled 'constant-multiple' but performs the differentiation of x to 1, which is a 'derivative' step. The label does not match the operation performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19
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.