Question

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

$f\left(x\right)=-4x+5$at$3$

Short answer

The derivative of given function at$x=3$ is $-4$.

Step-by-step solution

Step 1. Given Information 

We are given a function,

$f\left(x\right)=-4x+5$

We have to find the derivative of the given function at$3$.

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, $c=3$

$$\begin{gathered} f\left(x\right)=-4x+5 \\ f\left(3\right)=-4\left(3\right)+5 \\ =-12+5 \\ =-7 \end{gathered}$$

Step 3. Finding the derivative 

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

$$\begin{gathered} f\text{'}\left(3\right)=\underset{x\to 3}{\lim }\frac{f\left(x\right)-f\left(3\right)}{x-3} \\ =\underset{x\to 3}{\lim }\frac{-4x+5-(-7)}{x-3} \\ =\underset{x\to 3}{\lim }\frac{-4x+12}{x-3} \\ =\underset{x\to 3}{\lim }\frac{-4(x-3)}{x-3} \\ =\underset{x\to 3}{\lim }-4 \\ =-4 \end{gathered}$$

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