Derivative of \( \displaystyle \frac{3 x^{2} \ln{\left(x \right)}}{2} - \frac{3 x^{2}}{4} \)

Problem 2.1595 · hard Beautiful

1. \[ \frac{d}{d x} \left(\frac{3 x^{2} \ln{\left(x \right)}}{2} - \frac{3 x^{2}}{4}\right) \]
   algebraDifferentiate the function with respect to x. Distribute the constants.✓ Proved
2. \[ = - \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = - \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{3 \frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
   constant-multiplePull out the constant factors.✓ Proved
4. \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{3 \ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} - \frac{3 \frac{d}{d x} x^{2}}{4} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 3 x \ln{\left(x \right)} \]
   derivative simplify constant-multiple simplifyDifferentiate the individual components. Simplify the expression inside the parentheses. Distribute the constant 3/2. Combine like terms.✓ Proved