How do you figure out the derivative of sine without memorizing ...

[curlz]

I know the derivative of sin(x) = cos(x) but how do you figure that out?

[Millan]

use the lim definition of derivative,.....


lim  f(x+h) - f(x)
      _________
        delta x

as delta x approaches zero

real PITA but i vauguely remember it ,,,,,too hard to type in here.

[Millan]

You could use the definition of a derivative and solve it that way. The definition of a derivative states:
Derivative= lim(h-->0) [f(x + h) - f(x)] / h

So plug in sin(x) into that formula:
lim(h-->0) [sin(x + h) - sin(x)] / h

Use the sum and difference formula from to trig to expand the sin(x + h):
lim(h-->0) [sin(x)cos(h) + cos(x)sin(h) - sin(x)] / h

Now I'm just going to change the position of the things in the numerator:
lim(h-->0) [sin(x)cos(h) - sin(x) + cos(x) sin(h)] / h

Factor out a sin(x):
lim(h-->0) [sin(x)(cos(h) - 1) + cos(x)sin(h)] / h

Split up the numerator while keeping the same denominator:
lim [sin(x)(cos(h) - 1) / h] + lim[cos(x)sin(h) / h]

Now here's where some rules from limits come into play. Remember that the lim(x-->0) sinx / x = 1 and also the other rule to know is that lim(x-->0) cos(x)-1 / x = 0. What this means is that the cos(h)-1/h will drop out to 0 and that the sin(h)/h will drop out to 1. So you'll be left with:

lim[0] + lim(h-->0) cos(x)

 = cos(x)


hope it helps