Question

use the formula f(x) = lim_{z→x} \frac{f(z)-f(x)}{z - x} to find the derivative of the following function f(x)=\frac{5}{x + 4} f(x)=-\frac{5}{(x + 4)^2}

Explanation:

Step1: Find f(z)

$f(z)=\frac{5}{z + 4}$

Step2: Substitute f(z) and f(x) into the formula

$\lim_{z
ightarrow x}\frac{\frac{5}{z + 4}-\frac{5}{x + 4}}{z - x}=\lim_{z
ightarrow x}\frac{5(x + 4)-5(z + 4)}{(z - x)(z + 4)(x + 4)}$

Step3: Simplify the numerator

$\lim_{z
ightarrow x}\frac{5x+20 - 5z - 20}{(z - x)(z + 4)(x + 4)}=\lim_{z
ightarrow x}\frac{5(x - z)}{(z - x)(z + 4)(x + 4)}$

Step4: Cancel out terms

$\lim_{z
ightarrow x}\frac{- 5(z - x)}{(z - x)(z + 4)(x + 4)}=\lim_{z
ightarrow x}\frac{-5}{(z + 4)(x + 4)}$

Step5: Evaluate the limit

Substitute $z=x$ into $\frac{-5}{(z + 4)(x + 4)}$, we get $f'(x)=-\frac{5}{(x + 4)^2}$

Answer:

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