Question

use the formula f(x)=lim_{z→x} (f(z) - f(x))/(z - x) to find the derivative of the following function. f(x)=5/(x + 4) f(x)=□

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

\[

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

\]

Step3: Simplify the expression

Since \(x - z=-(z - x)\), we have:
\[

$$\begin{align*} f'(x)&=\lim_{z ightarrow x}\frac{-5(z - x)}{(z - x)(z + 4)(x + 4)}\\ &=\lim_{z ightarrow x}\frac{-5}{(z + 4)(x + 4)} \end{align*}$$

\]

Step4: Evaluate the limit

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

Answer:

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