Question

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

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

Short answer

Derivative of the function at$1$ is equal to$3$

Step-by-step solution

Step 1. Given information

Given function is$f\left(x\right)=x^3+x^2-2x$

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

Step 2. Definition of the derivative

Let's say a function$y=f\left(x\right)$

The derivative of the function at $c$is defined as

localid="1647083580386" $f\text{'}\left(c\right)=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

Step 3. Finding the derivative

By substituting the given function in the derivative definition,

we will get:

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

Step 4. Simplifying the limit

By rewriting the limit, we will have:

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