Derivative of \( \displaystyle \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \)

Problem 2.215 · hard Beautiful

1. \[ \frac{d}{d x} \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \]
   constant-multiplePull out the constant factor 1/4.✓ Proved
3. \[ = \frac{x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \frac{\left(2 \ln{\left(x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \frac{x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
   derivativeDifferentiate x**2.✓ Proved
5. \[ = \frac{x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(x \right)}\right)}{4} + \frac{x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
6. \[ = \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
   derivativeDifferentiate 2*log(x) and 1.✓ Proved
7. \[ = \frac{x \left(2 \ln{\left(x \right)} - 1\right)}{2} + \frac{x}{2} \]
   derivative algebraDifferentiate log(x). Simplify the term x**2 * (2/x).✓ Proved
8. \[ = x \ln{\left(x \right)} \]
   algebra algebra simplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved