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

Problem 2.2049 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - \ln{\left(\left(2 x\right)^{x^{2}} \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} \ln{\left(\left(2 x\right)^{x^{2}} \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} x^{2} \ln{\left(2 x \right)} \]
   rewriteRewrite the logarithm using the property log(a^b) = b*log(a).✓ Proved
4. \[ = - x^{2} \frac{d}{d x} \ln{\left(2 x \right)} - \ln{\left(2 x \right)} \frac{d}{d x} x^{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   productApply the product rule to the second term.✓ Proved
5. \[ = - x^{2} \frac{d}{d x} \ln{\left(2 x \right)} - 2 x \ln{\left(2 x \right)} + \frac{d}{d x} \frac{x^{2}}{2} \]
   powerDifferentiate x**2.✓ Proved
6. \[ = - 2 x \ln{\left(2 x \right)} - \frac{x \frac{d}{d x} 2 x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   chainApply the chain rule to log(2*x).✓ Proved
7. \[ = - 2 x \ln{\left(2 x \right)} - x + \frac{d}{d x} \frac{x^{2}}{2} \]
   derivative algebra algebraDifferentiate 2*x. Simplify the expression inside the parentheses. Simplify x**2/x.✓ Proved
8. \[ = - 2 x \ln{\left(2 x \right)} \]
   derivative algebra simplifyDifferentiate x**2/2. Distribute the negative sign. Combine like terms.✓ Proved