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

Problem 2.2057 · hard

1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + \ln{\left(\left(2 x\right)^{x^{2}} \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(\left(2 x\right)^{x^{2}} \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \ln{\left(2 x \right)} \]
   rewriteRewrite the logarithm using the property log(a^b) = b*log(a).✓ Proved
4. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right) \]
   algebraExpand the logarithm of the product.✓ Proved
5. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \ln{\left(2 \right)} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   sumApply the sum rule again.✓ Proved
6. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \ln{\left(2 \right)} \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   constant-multiplePull out the constant log(2).✓ Proved
7. \[ = - x + \ln{\left(2 \right)} \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   derivativeDifferentiate the first term.✓ Proved
8. \[ = - x + 2 x \ln{\left(2 \right)} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   derivativeDifferentiate the second term.✓ Proved
9. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} - x + 2 x \ln{\left(2 \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2} \]
   productApply the product rule to the third term.✓ Proved
10. \[ = 2 x \ln{\left(x \right)} + 2 x \ln{\left(2 \right)} \]
    derivative algebra algebra simplifyDifferentiate the components of the product rule. Simplify the term x**2/x. Distribute the terms. Combine like terms.✓ Proved
11. \[ = 2 x \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right) \]
    simplifyFactor out 2*x.✓ Proved
12. \[ = 2 x \ln{\left(2 x \right)} \]
    simplifyCombine the logarithms.✓ Proved