Derivative of \( \displaystyle \frac{e^{2 x}}{2 x} \)

Problem 2.372 · hard

1. \[ \frac{d}{d x} \frac{e^{2 x}}{2 x} \]
   rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = e^{2 x} \frac{d}{d x} \frac{1}{2 x} + \frac{\frac{d}{d x} e^{2 x}}{2 x} \]
   productApply the product rule.✓ Proved
3. \[ = e^{2 x} \frac{d}{d x} \frac{1}{2 x} + \frac{e^{2 x}}{x} \]
   chainDifferentiate the exponential term using the chain rule.✓ Proved
4. \[ = \frac{e^{2 x}}{x} - \frac{e^{2 x}}{2 x^{2}} \]
   chain algebra algebra simplifyDifferentiate the power term using the chain rule. Simplify the terms using exponent rules. Simplify the coefficients. Simplify the fraction.✓ Proved
5. \[ = \frac{2 x e^{2 x} - e^{2 x}}{2 x^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
6. \[ = \frac{\left(2 x - 1\right) e^{2 x}}{2 x^{2}} \]
   simplifyFactor out the exponential term.✓ Proved