Derivative of \( \displaystyle 5 x \ln{\left(x + 1 \right)} - 5 x + 5 \ln{\left(x + 1 \right)} \)

Problem 2.851 · hard Beautiful

1. \[ \frac{d}{d x} \left(5 x \ln{\left(x + 1 \right)} - 5 x + 5 \ln{\left(x + 1 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 5 \ln{\left(x + 1 \right)} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} 5 x \ln{\left(x + 1 \right)} + \frac{d}{d x} 5 \ln{\left(x + 1 \right)} - 5 \]
   constantDifferentiate the term 5*x.✓ Proved
4. \[ = 5 \frac{d}{d x} x \ln{\left(x + 1 \right)} + 5 \frac{d}{d x} \ln{\left(x + 1 \right)} - 5 \]
   constant-multipleFactor out the constant 5 from the terms.✓ Proved
5. \[ = 5 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 5 \ln{\left(x + 1 \right)} \frac{d}{d x} x + 5 \frac{d}{d x} \ln{\left(x + 1 \right)} - 5 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(x + 1 \right)} + 5 \ln{\left(x + 1 \right)} + 5 \frac{d}{d x} \ln{\left(x + 1 \right)} - 5 \]
   derivativeDifferentiate x.✓ Proved
7. \[ = \frac{5 x}{x + 1} + 5 \ln{\left(x + 1 \right)} - 5 + \frac{5}{x + 1} \]
   derivative algebraDifferentiate log(x + 1). Distribute the 5.✓ Proved
8. \[ = 5 \ln{\left(x + 1 \right)} - 5 + \frac{5 x + 5}{x + 1} \]
   algebra algebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
9. \[ = 5 \ln{\left(x + 1 \right)} \]
   simplify simplifySimplify the fraction. Final simplification.✓ Proved