Question

given $f(x)=\frac{8}{x}$, find $f(x)$ using the limit definition of the derivative.
$f(x)=$
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Explanation:

Step1: Recall limit - definition of derivative

The limit - definition of the derivative is $f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $f(x)=\frac{8}{x}$, then $f(x + h)=\frac{8}{x + h}$.

Step2: Substitute into the formula

\[

$$\begin{align*} f^{\prime}(x)&=\lim_{h ightarrow0}\frac{\frac{8}{x + h}-\frac{8}{x}}{h}\\ &=\lim_{h ightarrow0}\frac{\frac{8x-8(x + h)}{x(x + h)}}{h}\\ &=\lim_{h ightarrow0}\frac{8x-8x-8h}{hx(x + h)}\\ &=\lim_{h ightarrow0}\frac{- 8h}{hx(x + h)} \end{align*}$$

\]

Step3: Simplify the expression

Cancel out the $h$ terms: $\lim_{h
ightarrow0}\frac{-8}{x(x + h)}$.

Step4: Evaluate the limit

As $h
ightarrow0$, we get $f^{\prime}(x)=-\frac{8}{x^{2}}$.

Answer:

$-\frac{8}{x^{2}}$