Derivative of \( \displaystyle \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \)

Problem 2.1143 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(2 x - \ln{\left(4 x^{2} \right)}\right) \]
   algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(2*x)) to 2*x.✓ Proved
3. \[ = \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
   sumSplit the derivative using the difference rule.✓ Proved
4. \[ = 2 - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
   constantThe derivative of 2*x is 2.✓ Proved
5. \[ = 2 - \frac{d}{d x} \left(\ln{\left(x^{2} \right)} + \ln{\left(4 \right)}\right) \]
   algebraUse the property log(ab) = log(a) + log(b).✓ Proved
6. \[ = - \frac{d}{d x} \ln{\left(4 \right)} - \frac{d}{d x} \ln{\left(x^{2} \right)} + 2 \]
   sumDistribute the derivative.✓ Proved
7. \[ = 2 - \frac{d}{d x} \ln{\left(x^{2} \right)} \]
   constantThe derivative of the constant log(4) is 0.✓ Proved
8. \[ = 2 - \frac{d}{d x} 2 \ln{\left(x \right)} \]
   rewriteUse the power rule for logarithms: log(x**2) = 2*log(x).✓ Proved
9. \[ = 2 - 2 \frac{d}{d x} \ln{\left(x \right)} \]
   constant-multiplePull the constant factor 2 out of the derivative.✓ Proved
10. \[ = 2 - \frac{2}{x} \]
    derivative simplifyThe derivative of log(x) is 1/x. Simplify the expression.✓ Proved