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

Problem 2.1783 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{x^{2} \ln{\left(x \right)}}{2} + \frac{x^{2}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2} \ln{\left(x \right)}}{2}\right) \]
   sum constant-multipleApply the sum rule. Pull out the constant factors.✓ Proved
3. \[ = \frac{\frac{d}{d x} x^{2}}{4} - \frac{\frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
   constant-multipleSeparate the constants from the derivatives.✓ Proved
4. \[ = - \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} - \frac{\ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} + \frac{\frac{d}{d x} x^{2}}{4} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = - x \ln{\left(x \right)} \]
   derivative algebra algebra simplifyDifferentiate the individual components. Simplify the terms inside the parentheses and the second term. Distribute the constant -1/2. Combine like terms.✓ Proved