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

Problem 2.325 · hard

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