Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = (x² - 7x - 8) / (x + 1)

Verified step by step guidance

1. Step 1: Identify the function f(x) = \frac{x^2 - 7x - 8}{x + 1}. This is a rational function, so we will use the quotient rule to find the derivatives.
2. Step 2: Recall the quotient rule for derivatives: if you have a function \frac{u(x)}{v(x)}, then its derivative is given by \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}. Here, u(x) = x^2 - 7x - 8 and v(x) = x + 1.
3. Step 3: Compute the first derivative f'(x) using the quotient rule. First, find the derivatives of the numerator and denominator: u'(x) = 2x - 7 and v'(x) = 1. Then apply the quotient rule: f'(x) = \frac{(2x - 7)(x + 1) - (x^2 - 7x - 8)(1)}{(x + 1)^2}.
4. Step 4: Simplify the expression for f'(x) by expanding and combining like terms in the numerator. This will give you a simplified form of the first derivative.
5. Step 5: To find the second derivative f''(x), differentiate f'(x) using the quotient rule again. Finally, differentiate f''(x) to find the third derivative f'''(x), ensuring to simplify at each step.