Q. 3 - Calculus

Question

Calculate the derivative of $f\left(x\right)=x^2$ at  $c=4$.

Step-by-Step Solution

Answer

The derivative is $8$.

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, 

$$\begin{gathered} f^{\text{'}}\left(c\right)=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ {f}^{\text{'}}\left(4\right)=\underset{h\to 0}{\lim }\frac{(4+h{)}^2-(4{)}^2}{h} \\ =\underset{h\to 0}{\lim }\frac{16+h^2+8h-16}{h} \\ =\underset{h\to 0}{\lim }\frac{h(h+8)}{h} \\ =\underset{h\to 0}{\lim }(h+8) \\ =8 \end{gathered}$$