Question

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

$f\left(x\right)=2x^2+3x$at$1$

Short answer

The derivative of given function at $1$is $7$.

Step-by-step solution

Step 1. Given Information  

We are given a function,

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

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, $c=1$

$$\begin{gathered} f\left(x\right)=2x^2+3x \\ f\left(1\right)=2(1{)}^2+3(1) \\ =2+3 \\ =5 \end{gathered}$$

Step 3. Finding the derivative 

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

$$\begin{gathered} f\text{'}\left(1\right)=\underset{x\to 1}{\lim }\frac{f\left(x\right)-f\left(1\right)}{x-1} \\ =\underset{x\to 1}{\lim }\frac{2x^2+3x-5}{x-1} \\ =\underset{x\to 1}{\lim }\frac{(2x+5)(x-1)}{x-1} \\ =\underset{x\to 1}{\lim }(2x+5) \\ =2\left(1\right)+5 \\ =7 \end{gathered}$$

Hence, the derivative of the function at$1$ is $7$.