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

Problem 2.1923 · hard Beautiful

1. \[ \frac{d}{d x} \left(x^{2} \ln{\left(x \right)} - \frac{x^{2}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = - \frac{d}{d x} 0.5 x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   constantRewrite the fraction as a constant multiple.✓ Proved
4. \[ = - 0.5 \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
   constant-multiplePull out the constant factor.✓ Proved
5. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2} - 0.5 \frac{d}{d x} x^{2} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 2 x \ln{\left(x \right)} \]
   derivative algebra simplifyDifferentiate each part. Simplify the terms. Combine like terms.✓ Proved