Question

the derivative as a function
use limits to find f(x) if f(x)= - 2x.
f(x)=□

Explanation:

Step1: Recall derivative - limit formula

The derivative of a function $f(x)$ is defined as $f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $f(x)=-2x$, then $f(x + h)=-2(x + h)=-2x-2h$.

Step2: Substitute into the formula

Substitute $f(x + h)$ and $f(x)$ into the derivative - limit formula:
\[

$$\begin{align*} f^{\prime}(x)&=\lim_{h ightarrow0}\frac{(-2x-2h)-(-2x)}{h}\\ &=\lim_{h ightarrow0}\frac{-2x-2h + 2x}{h}\\ &=\lim_{h ightarrow0}\frac{-2h}{h} \end{align*}$$

\]

Step3: Simplify the limit

Cancel out the $h$ terms in the fraction: $\lim_{h
ightarrow0}\frac{-2h}{h}=\lim_{h
ightarrow0}-2$. Since the function $y = - 2$ is a constant function with respect to $h$, the limit as $h$ approaches $0$ is $-2$. So $f^{\prime}(x)=-2$.

Answer:

$-2$