Question

In Problems 21–32, find the derivative of each function at the given number.

$f\left(x\right)=x^2-3$at$0$

Short answer

The derivative of given function at $0$is $0$.

Step-by-step solution

Step 1. Given Information  

We are given a function,

$f\left(x\right)=x^2-3$

We have to find the derivative of the given function at $0$.

Step 2. Definition of derivative  

The derivative of the function at c is defined as,

$f\text{'}\left(c\right)=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

Here, $c=0$

$$\begin{gathered} f\left(x\right)=x^2-3 \\ f\left(0\right)=(0{)}^2-3 \\ =-3 \end{gathered}$$

Step 3. Finding the derivative 

By substituting the given function in the derivative definition, we get

$$\begin{gathered} f\text{'}\left(0\right)=\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0} \\ =\underset{x\to 0}{\lim }\frac{x^2-3-(-3)}{x} \\ =\underset{x\to 0}{\lim }\frac{x^2}{x} \\ =\underset{x\to 0}{\lim }x \\ =0 \end{gathered}$$

Hence, the derivative of the function at$0$ is $0$.