Q. 2 - Calculus

Question

Calculate the derivative of $f\left(x\right)=x^2$ at  $c=2$.

Step-by-Step Solution

Answer

The derivative is $4$.

Step 1. Given Information.

We are given a function, 

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

Step 2. Finding the derivative.

The derivative is given by, 

$$\begin{gathered} f^{\text{'}}\left(c\right)=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ {f}^{\text{'}}\left(2\right)=\underset{h\to 0}{\lim }\frac{(2+h{)}^2-(2{)}^2}{h} \\ =\underset{h\to 0}{\lim }\frac{4+h^2+4h-4}{h} \\ =\underset{h\to 0}{\lim }\frac{h(h+4)}{h} \\ =\underset{h\to 0}{\lim }(h+4) \\ =4 \end{gathered}$$