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

Problem 2.456 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(2 x \right)}}{4 x^{2}} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(2 x \right)}}{4 x^{2}} - \frac{\ln{\left(2 x \right)} \frac{d}{d x} 4 x^{2}}{16 x^{4}} \]
   quotientApply the quotient rule.✓ Proved
3. \[ = - \frac{\ln{\left(2 x \right)}}{2 x^{3}} + \frac{1}{4 x^{3}} \]
   chain algebra algebra algebraDifferentiate the numerator using the chain rule. Simplify the numerator of the first term. Simplify the first term. Simplify the second term.✓ Proved
4. \[ = \frac{1 - 2 \ln{\left(2 x \right)}}{4 x^{3}} \]
   algebraCombine the terms over a common denominator.✓ Proved
5. \[ = \frac{- 2 \ln{\left(x \right)} - 2 \ln{\left(2 \right)} + 1}{4 x^{3}} \]
   algebra algebraExpand the logarithm. Distribute the constant.✓ Proved
6. \[ = \frac{1 - 2 \ln{\left(2 x \right)}}{4 x^{3}} \]
   simplifyFinal simplified form.✓ Proved