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

Problem 2.722 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{2 x}}{2 x} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(2 x - \ln{\left(2 x \right)}\right) \]
   algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(2*x)) to 2*x.✓ Proved
3. \[ = \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(2 x \right)} \]
   sumApply the difference rule for derivatives.✓ Proved
4. \[ = 2 - \frac{d}{d x} \ln{\left(2 x \right)} \]
   constantThe derivative of 2*x is 2.✓ Proved
5. \[ = 2 - \frac{\frac{d}{d x} 2 x}{2 x} \]
   chainApply the chain rule to log(2*x).✓ Proved
6. \[ = 2 - \frac{1}{x} \]
   derivative simplifyThe derivative of 2*x is 2. Simplify the expression.✓ Proved