Derivative of \( \displaystyle 2 x \ln{\left(x - 3 \right)} - 2 x - 6 \ln{\left(x - 3 \right)} \)

Problem 2.1442 · hard Beautiful

1. \[ \frac{d}{d x} \left(2 x \ln{\left(x - 3 \right)} - 2 x - 6 \ln{\left(x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 6 \ln{\left(x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} 2 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 6 \ln{\left(x - 3 \right)} - 2 \]
   constantDifferentiate the linear term.✓ Proved
4. \[ = 2 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 2 \ln{\left(x - 3 \right)} - \frac{d}{d x} 6 \ln{\left(x - 3 \right)} - 2 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = \frac{2 x}{x - 3} + 2 \ln{\left(x - 3 \right)} - \frac{d}{d x} 6 \ln{\left(x - 3 \right)} - 2 \]
   logarithmicDifferentiate the natural logarithm.✓ Proved
6. \[ = \frac{2 x}{x - 3} + 2 \ln{\left(x - 3 \right)} - 2 - \frac{6}{x - 3} \]
   constant-multipleDifferentiate the constant multiple of the logarithm.✓ Proved
7. \[ = 2 \ln{\left(x - 3 \right)} - 2 + \frac{2 x - 6}{x - 3} \]
   algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
8. \[ = 2 \ln{\left(x - 3 \right)} \]
   algebra simplifySimplify the fraction. Simplify the remaining terms.✓ Proved