Derivative of \( \displaystyle - 5 x + \left(5 x - 15\right) \ln{\left(x - 3 \right)} \)

Problem 2.2132 · hard Beautiful

1. \[ \frac{d}{d x} \left(- 5 x + \left(5 x - 15\right) \ln{\left(x - 3 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x - 15\right) \ln{\left(x - 3 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x \ln{\left(x - 3 \right)} - 15 \ln{\left(x - 3 \right)}\right) \]
   algebraDistribute the term (5*x - 15).✓ Proved
4. \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} \]
   sumApply the sum rule to the second part.✓ Proved
5. \[ = \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
   constantDifferentiate the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} 5 x - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
   productApply the product rule to the second term.✓ Proved
7. \[ = 5 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 5 \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
   derivativeDifferentiate 5*x.✓ Proved
8. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
   logarithmicDifferentiate the natural logarithm.✓ Proved
9. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - 5 - \frac{15}{x - 3} \]
   derivative algebraDifferentiate the third term. Remove parentheses.✓ Proved
10. \[ = 5 \ln{\left(x - 3 \right)} - 5 + \frac{5 x - 15}{x - 3} \]
    algebra algebraCombine the fractions. Factor the numerator.✓ Proved
11. \[ = 5 \ln{\left(x - 3 \right)} \]
    algebra simplifySimplify the fraction. Combine the constant terms.✓ Proved