Derivative of \( \displaystyle 3 x \ln{\left(x - 3 \right)} - 3 x - 9 \ln{\left(x - 3 \right)} \)

Problem 2.1394 · hard Beautiful

1. \[ \frac{d}{d x} \left(3 x \ln{\left(x - 3 \right)} - 3 x - 9 \ln{\left(x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 9 \ln{\left(x - 3 \right)} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = \frac{d}{d x} 3 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 9 \ln{\left(x - 3 \right)} - 3 \]
   constantDifferentiate the linear term 3*x.✓ Proved
4. \[ = 3 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 3 \ln{\left(x - 3 \right)} - \frac{d}{d x} 9 \ln{\left(x - 3 \right)} - 3 \]
   productApply the product rule to 3*x*log(x - 3).✓ Proved
5. \[ = \frac{3 x}{x - 3} + 3 \ln{\left(x - 3 \right)} - 3 - \frac{9}{x - 3} \]
   logarithmic constant-multipleDifferentiate the natural logarithm term. Distribute the constant 3 into the derivative term.✓ Proved
6. \[ = 3 \ln{\left(x - 3 \right)} - 3 + \frac{3 x - 9}{x - 3} \]
   algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
7. \[ = 3 \ln{\left(x - 3 \right)} \]
   simplify simplifySimplify the fraction. Final simplification.✓ Proved