Question

Use the definition of the derivative to find f' for each function f.

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

Short answer

$f\text{'}\left(x\right)=$$2x$

Step-by-step solution

Step 1. Given information. 

We have been given a function:

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

We have to find f' using the definition of the derivative.

Step 2. Find the derivative 

Derivative of the function can be found as :

$$\begin{gathered} \begin{array}{r}\underset{h\to 0}{lim} \frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim} \frac{4+(x+h{)}^2-\left(4+x^2\right)}{h}\\ =\underset{h\to 0}{lim} \frac{4+\left(x^2+2xh+h^2\right)-4-x^2}{h}\\ =\underset{h\to 0}{lim} \frac{h(2x+h)}{h}\\ =\underset{h\to 0}{lim} 2x+h\end{array} \\ =2x \end{gathered}$$