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

Problem 2.1573 · hard Beautiful

1. \[ \frac{d}{d x} \left(2 x \ln{\left(x \right)} - 2 x\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 2 x \ln{\left(x \right)} - 2 \]
   constantThe derivative of 2*x is 2.✓ Proved
4. \[ = 2 \frac{d}{d x} x \ln{\left(x \right)} - 2 \]
   constant-multipleFactor out the constant 2.✓ Proved
5. \[ = 2 x \frac{d}{d x} \ln{\left(x \right)} + 2 \ln{\left(x \right)} \frac{d}{d x} x - 2 \]
   productApply the product rule to x*log(x).✓ Proved
6. \[ = 2 \ln{\left(x \right)} \]
   derivative algebra algebra simplifyDifferentiate x and log(x). Simplify the expression inside the parentheses. Distribute the 2. Combine like terms.✓ Proved