Derivative of \( \displaystyle \left(x - 1\right)^{\frac{1}{x - 1}} \)

Problem 2.378 · hard

1. \[ \frac{d}{d x} \left(x - 1\right)^{\frac{1}{x - 1}} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}} \]
   rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
3. \[ = e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}} \frac{d}{d x} \frac{\ln{\left(x - 1 \right)}}{x - 1} \]
   chainApply the chain rule.✓ Proved
4. \[ = \left(\ln{\left(x - 1 \right)} \frac{d}{d x} \frac{1}{x - 1} + \frac{\frac{d}{d x} \ln{\left(x - 1 \right)}}{x - 1}\right) e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}} \]
   productApply the product rule to the inner expression.✓ Proved
5. \[ = \left(\ln{\left(x - 1 \right)} \frac{d}{d x} \frac{1}{x - 1} + \frac{1}{\left(x - 1\right)^{2}}\right) e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}} \]
   derivativeDifferentiate the logarithm term.✓ Proved
6. \[ = \left(- \frac{\ln{\left(x - 1 \right)}}{\left(x - 1\right)^{2}} + \frac{1}{\left(x - 1\right)^{2}}\right) e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}} \]
   derivativeDifferentiate the reciprocal term.✓ Proved
7. \[ = \frac{\left(1 - \ln{\left(x - 1 \right)}\right) e^{\frac{\ln{\left(x - 1 \right)}}{x - 1}}}{\left(x - 1\right)^{2}} \]
   algebraFactor out the common term.✓ Proved
8. \[ = \frac{\left(1 - \ln{\left(x - 1 \right)}\right) \left(x - 1\right)^{\frac{1}{x - 1}}}{\left(x - 1\right)^{2}} \]
   simplifyConvert the exponential back to the original power form.≈ Checked numerically