Q. 41 - Calculus

Question

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

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

Step-by-Step Solution

Answer

$f\text{'}\left(x\right)=3x^2$

Step 1. Given information.

We have been given a function:

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

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

Step 2. Find the derivative

Derivative of the function can be found as :  

$$\begin{gathered} \begin{array}{r}\underset{h\to 0}{lim} \frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim} \frac{(x+h{)}^2+2-\left(x^3+2\right)}{h}\\ =\underset{h\to 0}{lim} \frac{\left(x^3+3x^{2}h+3x{h}^2+h^3\right)+2-x^3-2}{h}\\ =\underset{h\to 0}{lim} \frac{h\left(3x^2+3xh+h^2\right)}{h}\\ =\underset{h\to 0}{lim} 3x^2+3xh+h^2\end{array} \\ =3x^2 \end{gathered}$$