use the formula f(x)=lim_{z→x} (f(z) - f(x))/(z - x) to find the derivative of the following function…

Explanation:

Step1: Find f(z)

Given (f(x)=\frac{5}{x + 4}), then (f(z)=\frac{5}{z + 4}).

Step2: Substitute into the derivative formula

[ \begin{align*} f'(x)&=\lim_{z\rightarrow x}\frac{f(z)-f(x)}{z - x}\ &=\lim_{z\rightarrow x}\frac{\frac{5}{z + 4}-\frac{5}{x + 4}}{z - x}\ &=\lim_{z\rightarrow x}\frac{5(x + 4)-5(z + 4)}{(z - x)(z + 4)(x + 4)}\ &=\lim_{z\rightarrow x}\frac{5x+20 - 5z-20}{(z - x)(z + 4)(x + 4)}\ &=\lim_{z\rightarrow x}\frac{5(x - z)}{(z - x)(z + 4)(x + 4)} \end{align*} ]

Step3: Simplify the expression

Since (x - z=-(z - x)), we have: [ \begin{align*} f'(x)&=\lim_{z\rightarrow x}\frac{-5(z - x)}{(z - x)(z + 4)(x + 4)}\ &=\lim_{z\rightarrow x}\frac{-5}{(z + 4)(x + 4)} \end{align*} ]

Step4: Evaluate the limit

Substitute (z = x) into the expression: [f'(x)=\frac{-5}{(x + 4)(x + 4)}=-\frac{5}{(x + 4)^2}]

Answer:

(-\frac{5}{(x + 4)^2})