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

$$
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$$

Step3: Simplify the expression

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

$$
LATEXBLOCK1
$$

Step4: Evaluate the limit

Substitute \(z = x\) into the expression:

$$f'(x)=\frac{-5}{(x + 4)(x + 4)}=-\frac{5}{(x + 4)^2}$$

Answer:

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