Question

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

$f\left(x\right)=2x^2+1$at$-1$

Short answer

The derivative of given function at $-1$is $-4$.

Step-by-step solution

Step 1. Given Information  

We are given a function,

$f\left(x\right)=2x^2+1$

We have to find the derivative of the given function at $-1$.

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, $c=-1$

$$\begin{gathered} f\left(x\right)=2x^2+1 \\ f(-1)=2(-1{)}^2+1 \\ =2+1 \\ =3 \end{gathered}$$

Step 3. Finding the derivative 

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

$$\begin{gathered} f\text{'}(-1)=\underset{x\to -1}{\lim }\frac{f\left(x\right)-f(-1)}{x-(-1)} \\ =\underset{x\to -1}{\lim }\frac{2x^2+1-3}{x+1} \\ =\underset{x,1}{\lim }\frac{2x^2-2}{x+1} \\ =\underset{x\to -1}{\lim }\frac{2\left(x^2-1\right)}{x+1} \\ =\underset{x\to -1}{\lim }\frac{2(x+1)(x-1)}{x+1} \\ =\underset{x\to -1}{\lim }2(x-1) \\ =2(-1-1) \\ =-4 \end{gathered}$$

Hence, the derivative of the function at$-1$ is $-4$.