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

Problem 2.1505 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(x \right)}^{2} - 2 x \ln{\left(x \right)} + 2 x\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} 2 x - \frac{d}{d x} 2 x \ln{\left(x \right)} + \frac{d}{d x} x \ln{\left(x \right)}^{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 2 x - 2 \frac{d}{d x} x \ln{\left(x \right)} + \frac{d}{d x} x \ln{\left(x \right)}^{2} \]
   constant-multipleFactor out the constant 2.✓ Proved
4. \[ = - 2 \frac{d}{d x} x \ln{\left(x \right)} + \frac{d}{d x} x \ln{\left(x \right)}^{2} + 2 \]
   constantThe derivative of 2*x is 2.✓ Proved
5. \[ = - 2 x \frac{d}{d x} \ln{\left(x \right)} + x \frac{d}{d x} \ln{\left(x \right)}^{2} + \ln{\left(x \right)}^{2} \frac{d}{d x} x - 2 \ln{\left(x \right)} \frac{d}{d x} x + 2 \]
   productApply the product rule to each term.✓ Proved
6. \[ = - 2 x \frac{d}{d x} \ln{\left(x \right)} + x \frac{d}{d x} \ln{\left(x \right)}^{2} + \ln{\left(x \right)}^{2} - 2 \ln{\left(x \right)} + 2 \]
   constant algebra algebraThe derivative of x is 1. Simplify the expression. Distribute the -2.✓ Proved
7. \[ = 2 x \ln{\left(x \right)} \frac{d}{d x} \ln{\left(x \right)} - 2 x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)}^{2} - 2 \ln{\left(x \right)} + 2 \]
   chainApply the chain rule to log(x)**2.✓ Proved
8. \[ = \ln{\left(x \right)}^{2} \]
   logarithmic algebra algebra algebra algebra simplifyThe derivative of log(x) is 1/x. Rewrite the fraction. Simplify x/x. Combine the log(x) terms. Simplify 2*x*(1/x). Final simplification.✓ Proved