find f(x) for the given function. f(x) = 4 / (8 - x) f(x) =

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{4}{8 - x}$ as $f(x)=4(8 - x)^{-1}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 8 - x$, so $y = 4u^{-1}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$. $\frac{dy}{du}=-4u^{-2}=-\frac{4}{u^{2}}$ and $\frac{du}{dx}=-1$.

Step3: Calculate $f^\prime(x)$

By the chain - rule $f^\prime(x)=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 8 - x$ back in: $f^\prime(x)=(-\frac{4}{(8 - x)^{2}})\times(-1)=\frac{4}{(8 - x)^{2}}$.

Answer:

$\frac{4}{(8 - x)^{2}}$