Question

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

$f\left(x\right)=3x^2-4x$at$2$

Short answer

The derivative of given function at $2$is $8$.

Step-by-step solution

Step 1. Given Information  

We are given a function,

$f\left(x\right)=3x^2-4x$

We have to find the derivative of the given function at $2$.

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, $c=2$

$$\begin{gathered} f\left(x\right)=3x^2-4x \\ f\left(2\right)=3(2{)}^2-4(2) \\ =12-8 \\ =4 \end{gathered}$$

Step 3. Finding the derivative 

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

$$\begin{gathered} f\text{'}\left(2\right)=\underset{x\to 2}{\lim }\frac{f\left(x\right)-f\left(2\right)}{x-2} \\ =\underset{x\to 2}{\lim }\frac{3x^2-4x-4}{x-2} \\ =\underset{x\to 2}{\lim }\frac{(3x+2)(x-2)}{x-2} \\ =\underset{x\to 2}{\lim }(3x+2) \\ =3\left(2\right)+2 \\ =6+2 \\ =8 \end{gathered}$$

Hence, the derivative of the function at$2$ is $8$.