Derivative of \( \displaystyle \left(1 + \frac{1}{x - 3}\right)^{x - 3} \)
Problem 2.502 · hard
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{x - 3}\right)^{x - 3} \).
- \[ \frac{d}{d x} \left(1 + \frac{1}{x - 3}\right)^{x - 3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]rewriteRewrite the base and exponent using the exponential identity.≈ Checked numerically
- \[ = e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \frac{d}{d x} \left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(1 + \frac{1}{x - 3} \right)} + \ln{\left(1 + \frac{1}{x - 3} \right)} \frac{d}{d x} \left(x - 3\right)\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(1 + \frac{1}{x - 3} \right)} + \ln{\left(1 + \frac{1}{x - 3} \right)}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]derivativeDifferentiate the first factor of the product.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x - 3} \right)} + \frac{\left(x - 3\right) \frac{d}{d x} \left(1 + \frac{1}{x - 3}\right)}{1 + \frac{1}{x - 3}}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x - 3} \right)} + \frac{\left(x - 3\right) \left(\frac{d}{d x} 1 + \frac{d}{d x} \frac{1}{x - 3}\right)}{1 + \frac{1}{x - 3}}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x - 3} \right)} + \frac{\left(x - 3\right) \frac{d}{d x} \frac{1}{x - 3}}{1 + \frac{1}{x - 3}}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]constantThe derivative of the constant 1 is 0.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x - 3} \right)} - \frac{1}{\left(1 + \frac{1}{x - 3}\right) \left(x - 3\right)}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]powerDifferentiate the reciprocal term using the power rule.✓ Proved
- \[ = \left(\ln{\left(1 + \frac{1}{x - 3} \right)} - \frac{1}{x - 2}\right) e^{\left(x - 3\right) \ln{\left(1 + \frac{1}{x - 3} \right)}} \]algebra algebra algebra algebraSimplify the fraction inside the expression. Simplify the reciprocal of the fraction. Cancel the common term (x - 3) in the numerator and denominator. Simplify the expression inside the parenthesis.✓ Proved
- \[ = \left(1 + \frac{1}{x - 3}\right)^{x - 3} \left(\ln{\left(1 + \frac{1}{x - 3} \right)} - \frac{1}{x - 2}\right) \]simplifyConvert back to the original base-exponent form.≈ Checked numerically
Answer \( \frac{\left(\frac{x - 2}{x - 3}\right)^{x - 3} \left(\left(x - 2\right) \log{\left(\frac{x - 2}{x - 3} \right)} - 1\right)}{x - 2} \)
✓ Nihil obstat Lines: 13 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 - 2)/(x - 3))**(x - 3)*((x - 2)*log((x - 2)/(x - 3)) - 1) + (-(x - 2)*log((x - 2)/(x - 3)) + 1)*exp((x - 3)*log((x - 2)/(x - 3))))/(x - 2); numeric agreement only, at 24 of 24 sampled points undefined where x - 3 = 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 - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0 undefined where 1 + 1/(x - 3) = 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 - 3) = 0 undefined where x - 3 = 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 - 3) = 0 undefined where x - 3 = 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 - 3) = 0 undefined where x - 3 = 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 - 3) = 0 undefined where x - 3 = 0 undefined where x - 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 2 = 0 undefined where x - 3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 2 = 0 undefined where x - 3 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 2 = 0 undefined where x - 3 = 0 |
| 14 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (((x - 2)/(x - 3))**(x - 3)*(-(x - 2)*log((x - 2)/(x - 3)) + 1) + ((x - 2)*log((x - 2)/(x - 3)) - 1)*exp((x - 3)*log((x - 2)/(x - 3))))/(x - 2); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x - 2 = 0 undefined where x - 3 = 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 - 2 = 0 undefined where x - 3 = 0 |
| 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 chain rule, product rule, and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, product rule, and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20
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.