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

Problem 2.340 · hard

1. \[ \frac{d}{d x} \frac{e^{x - 1}}{x} \]
   rewriteStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
2. \[ = e^{x - 1} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} e^{x - 1}}{x} \]
   productApply the product rule.✓ Proved
3. \[ = e^{x - 1} \frac{d}{d x} \frac{1}{x} + \frac{e^{x - 1}}{x} \]
   derivativeDifferentiate the first term.✓ Proved
4. \[ = \frac{e^{x - 1}}{x} - \frac{e^{x - 1}}{x^{2}} \]
   derivative algebraDifferentiate the second term. Simplify the signs.✓ Proved
5. \[ = \left(\frac{1}{x} - \frac{1}{x^{2}}\right) e^{x - 1} \]
   algebraFactor out the common term.✓ Proved
6. \[ = \frac{\left(x - 1\right) e^{x - 1}}{x^{2}} \]
   simplifyCombine the terms into a single fraction.✓ Proved