Question

Derivative this using the limit definition.

\[f(x) = sinx\]

Answer 1

I'm currently stuck on this: \[\lim_{\Delta x \rightarrow 0} \frac{ \cos x \sin \Delta x + \cos \Delta x \sin x - \sin x }{ \Delta x }\]

Answer 2

What to do?

Answer 3

factor the \(\sin x\) from the 2nd and 3rd summands of the numerator

Answer 4

you need to know something first, namely that
\[\lim_{x\to 0}\frac{\sin(x)}{x}=1\] and also that
\[\lim_{x\to 0}\frac{\cos(x)-1}{x}=0\]

Answer 5

i would look in any text to see it worked out, although i guess someone could write it here

Answer 6

oh also you need to know that
\[\sin(x+h)=\sin(x)\cos(h)+\cos(x)\sin(h)\] via the "addition angle" formula

Answer 7

so that
\[\sin(x+h)-\sin(x)=\sin(x+h)=\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)\]

Answer 8

Getting through it ok micah? requires a little bit of factoring from the part you were stuck on

Answer 9

I got it. Thanks everyone, especially satellite73.\[\lim_{\Delta x \rightarrow 0} \frac{cosxsin \Delta x}{\Delta x} + \frac{\sin x(\cos \Delta x - 1)}{\Delta x}= \lim_{\Delta x \rightarrow 0} 1\cos x + 0\sin x = \cos x\]
Saying, it's interesting how deriviatives work, lol.