Derivative of \( \displaystyle \left(x - 2\right)^{x - 3} \)

Problem 2.461 · medium

1. \[ \frac{d}{d x} \left(x - 2\right)^{x - 3} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} e^{\left(x - 3\right) \ln{\left(x - 2 \right)}} \]
   rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
3. \[ = e^{\left(x - 3\right) \ln{\left(x - 2 \right)}} \frac{d}{d x} \left(x - 3\right) \ln{\left(x - 2 \right)} \]
   chainApply the chain rule.✓ Proved
4. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(x - 2 \right)} + \ln{\left(x - 2 \right)} \frac{d}{d x} \left(x - 3\right)\right) e^{\left(x - 3\right) \ln{\left(x - 2 \right)}} \]
   productApply the product rule to the inner expression.✓ Proved
5. \[ = \left(\left(x - 3\right) \frac{d}{d x} \ln{\left(x - 2 \right)} + \ln{\left(x - 2 \right)}\right) e^{\left(x - 3\right) \ln{\left(x - 2 \right)}} \]
   derivativeDifferentiate the first part of the product.✓ Proved
6. \[ = \left(\frac{x - 3}{x - 2} + \ln{\left(x - 2 \right)}\right) e^{\left(x - 3\right) \ln{\left(x - 2 \right)}} \]
   derivativeDifferentiate the second part of the product.✓ Proved
7. \[ = \left(x - 2\right)^{x - 3} \left(\frac{x - 3}{x - 2} + \ln{\left(x - 2 \right)}\right) \]
   simplifyConvert the exponential back to its original form.≈ Checked numerically
8. \[ = \left(x - 2\right)^{x - 3} \left(\ln{\left(x - 2 \right)} + 1 - \frac{1}{x - 2}\right) \]
   algebraSplit the fraction inside the parentheses.✓ Proved
9. \[ = \left(x - 2\right)^{x - 3} \left(\frac{x - 3}{x - 2} + \ln{\left(x - 2 \right)}\right) \]
   simplifyKeep the fraction form for the final answer.✓ Proved