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

Problem 2.614 · hard

1. \[ \frac{d}{d x} \left(x - 3\right)^{2} \ln{\left(x - 3 \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \left(x - 3\right)^{2} \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)^{2} \]
   productApply the product rule.✓ Proved
3. \[ = \left(x - 3\right)^{2} \frac{d}{d x} \ln{\left(x - 3 \right)} + \left(2 x - 6\right) \ln{\left(x - 3 \right)} \]
   powerDifferentiate the first part using the power rule.✓ Proved
4. \[ = x + \left(2 x - 6\right) \ln{\left(x - 3 \right)} - 3 \]
   logarithmic algebraDifferentiate the natural logarithm. Simplify the fraction by canceling (x - 3).✓ Proved
5. \[ = \left(x - 3\right) \left(2 \ln{\left(x - 3 \right)} + 1\right) \]
   algebraFactor out the common term (x - 3).✓ Proved