Questions: Consider the following. f(x) = x / (x - 8) Find the first derivative of the function. f'(x) = Find the second derivative of the function. f'''(x) = Consider the following. g(x) = 5e^x / x Find the first derivative of the function. g'(x) = Find the second derivative of the function. g*(x) =

Solution Steps

1. First Derivative of \( f(x) = \frac{x}{x-8} \): Use the quotient rule for derivatives, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \).

2. Second Derivative of \( f(x) \): Once you have the first derivative, apply the derivative rules again to find the second derivative.

3. First Derivative of \( g(x) = \frac{5e^x}{x} \): Again, use the quotient rule to find the derivative of this function.

To find the first derivative of \( f(x) \), we use the quotient rule. The quotient rule states that if \( f(x) = \frac{u}{v} \), then \( f'(x) = \frac{u'v - uv'}{v^2} \).

For \( f(x) = \frac{x}{x-8} \), let \( u = x \) and \( v = x-8 \). Then \( u' = 1 \) and \( v' = 1 \).

Applying the quotient rule:

\[ f'(x) = \frac{1 \cdot (x-8) - x \cdot 1}{(x-8)^2} = \frac{x-8-x}{(x-8)^2} = \frac{-8}{(x-8)^2} \]

To find the second derivative, we differentiate \( f'(x) = \frac{-8}{(x-8)^2} \) again with respect to \( x \).

Using the power rule and chain rule:

\[ f''(x) = \frac{d}{dx}\left(\frac{-8}{(x-8)^2}\right) = \frac{16}{(x-8)^3} \]

For \( g(x) = \frac{5e^x}{x} \), we again use the quotient rule. Let \( u = 5e^x \) and \( v = x \). Then \( u' = 5e^x \) and \( v' = 1 \).

Applying the quotient rule:

\[ g'(x) = \frac{5e^x \cdot x - 5e^x \cdot 1}{x^2} = \frac{5xe^x - 5e^x}{x^2} = \frac{5e^x(x-1)}{x^2} \]

Final Answer