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

Problem 2.1768 · hard Beautiful

1. \[ \frac{d}{d x} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = - \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
   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 term x**2/x. Distribute the constant 1/2. Combine like terms.✓ Proved