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

Problem 2.13 · hard

1. \[ \frac{d}{d x} x^{2} \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) + \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) + 2 x \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(- \frac{d}{d x} \frac{1}{2} + \frac{d}{d x} \ln{\left(2 x \right)}\right) + 2 x \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \]
   sumDistribute the derivative to the terms in the parentheses.✓ Proved
5. \[ = x^{2} \frac{d}{d x} \ln{\left(2 x \right)} + 2 x \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \]
   constant algebraThe derivative of the constant 1/2 is 0. Simplify the expression by removing the zero term.✓ Proved
6. \[ = 2 x \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) + \frac{x \frac{d}{d x} 2 x}{2} \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = 2 x \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) + x \]
   derivative algebra algebraDifferentiate the inner function 2*x. Simplify the product of 1/(2*x) and 2. Simplify the term x**2 * (1/x).✓ Proved
8. \[ = 2 x \ln{\left(2 x \right)} \]
   algebra simplifyDistribute 2*x into the parentheses. Combine like terms.✓ Proved