Q. 35 - Calculus

Question

Use (a) the h→0 definition of the derivative and then (b) the z→c definition of the derivative to find f'(c) for each function f and value x = c. 

$f\left(x\right)=\sin x,x=0$

Step-by-Step Solution

Answer

(a) $f^{\text{'}}\left(c\right)=1$

(b) $f^{\text{'}}\left(c\right)=1$

Part (a) Step 1. Given information.

Given function is $f\left(x\right)=\sin x$

We have to find $f^{\text{'}}\left(c\right)$

Part (a) Step 2. Find the f ' ( c )

We have to find the derivative of the function using h→0 definition,

Therefore,

$\begin{array}{r}\underset{h\to 0}{lim} \frac{f(0+h)-f\left(0\right)}{h}=\underset{h\to 0}{lim} \frac{\sin (0+h)-\sin 0}{h}\\ =\underset{h\to 0}{lim} \frac{\sin \left(h\right)}{h}\\ =1\end{array}$

Part (b) Step 1. Find f ' ( c )

Find the derivate of the function using x→0 definition
Therefore,

$\begin{array}{r}\underset{x\to 0}{lim} \frac{f\left(x\right)-f\left(0\right)}{x-0}-\underset{x\to 0}{lim} \frac{\sin x-\sin 0}{x}\\ =\underset{x\to 0}{lim} \frac{\sin x}{x}\\ =1\end{array}$