Derivative of y(x)=sin x: \cos x

[abstrakt!]

I am studying this from a book I found online, and I need a little bit of help.

Homework Statement

Find the derivative when $y(x)=\sin x$

The Attempt at a Solution

$\frac{dy}{dx} \ = \ limit \ of \ \frac {\Delta y}{\Delta x} \ = \ \lim h \rightarrow 0 \ \frac{\sin(x+h)-\sin x}{h}$

$\sin(x+h)=\sin x \cos h + \cos x \sin h$

$\frac{\Delta y}{\Delta x} \ = \ \frac {\sin x \cos h + \cos x \sin h-\sin x}{h} \ = \ \sin x ( \frac{\cos h-1}{h}) + \cos x (\frac{\sin h}{h})$

 

[rock.freak667]

now take the limit as h→0, what does sinh/h tend to? and what does (cosh-1)/h tend to?

 

[HallsofIvy]

Strictly speaking, how you do that depends upon what your definitions of "sine" and "cosine" are- and there are several possible. What definitions are you using?