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

Problem 2.338 · medium

1. \[ \frac{d}{d x} x^{x - 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} e^{\left(x - 1\right) \ln{\left(x \right)}} \]
   rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
3. \[ = e^{\left(x - 1\right) \ln{\left(x \right)}} \frac{d}{d x} \left(x - 1\right) \ln{\left(x \right)} \]
   chainApply the chain rule.✓ Proved
4. \[ = \left(\frac{d}{d x} x \ln{\left(x \right)} - \frac{d}{d x} \ln{\left(x \right)}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
   algebraDistribute the derivative over the subtraction.✓ Proved
5. \[ = \left(\frac{d}{d x} x \ln{\left(x \right)} - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
   derivativeDifferentiate the log(x) term.✓ Proved
6. \[ = \left(x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
   productApply the product rule to x * log(x).✓ Proved
7. \[ = \left(\ln{\left(x \right)} + 1 - \frac{1}{x}\right) e^{\left(x - 1\right) \ln{\left(x \right)}} \]
   derivative algebraDifferentiate the individual terms. Simplify the expression inside the parentheses.✓ Proved
8. \[ = x^{x - 1} \left(\ln{\left(x \right)} + 1 - \frac{1}{x}\right) \]
   simplifyConvert the exponential back to the original power form.≈ Checked numerically