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

Problem 2.2016 · hard Beautiful

1. \[ \frac{d}{d x} \left(- 5 x + \left(5 x + 5\right) \ln{\left(x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + 5\right) \ln{\left(x + 1 \right)} \]
   sum algebraApply the sum rule. Factor out the constant 5.✓ Proved
3. \[ = \frac{d}{d x} \left(- 5 x\right) + 5 \frac{d}{d x} \left(x + 1\right) \ln{\left(x + 1 \right)} \]
   constant-multipleApply the constant multiple rule.✓ Proved
4. \[ = 5 \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + 5 \ln{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + \frac{d}{d x} \left(- 5 x\right) \]
   productApply the product rule to the second term.✓ Proved
5. \[ = 5 \left(x + 1\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + 5 \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
6. \[ = 5 \ln{\left(x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + 5 \]
   logarithmic algebra algebraDifferentiate the natural logarithm. Simplify the fraction. Distribute the 5.✓ Proved
7. \[ = 5 \ln{\left(x + 1 \right)} \]
   derivative simplifyDifferentiate the first term. Combine the constant terms.✓ Proved