How to Find the Derivative of y=cosx Using the Limit Process?

[lilxchristina]

how do you find the derivative of y=cosx by using the limit process of

limit as h --> 0 is f(x+h) - f(x) / all over h.

i did this with y=sinx, and the answer was cosx, but I'm having trouble figuring out y=cosx.


help?

 

[arildno]

Well,
$$\frac{\cos(x+h)-cos(x)}{h}=\frac{\cos(h)-1}{h}\cos(x)-\frac{\sin(h)}{h}\sin(x)=-\frac{\sin(h)}{h}(\frac{\sin(h)}{\cos(h)+1}\cos(x)+\sin(x))$$
using well-known identities. Can you finish this?

 

[ZioX]

You need to know that

$$\lim_{t\to 0} \frac{\sin(t)}{t}=1.$$

The most common elementary proof of this is geometric.