How do you find the derivative of ##(cos x)## using the limit definition?

See the explanation section below.

We'll need the following facts:

From trigonometry: ##cos(A+B) = cosAcosB-sinAsinB##

Fundamental trigonometric limits:

##lim_(theta rarr0) sin theta /theta = 1##

##lim_(theta rarr0) (cos theta - 1) /theta = 0##

And here we go:

##f(x) = cosx##

##f'(x) = lim_(hrarr0)(cos(x+h)-cosx)/h##

## = lim_(hrarr0)(cosxcosh-sinxsinh-cosx)/h##

## = lim_(hrarr0)(cosxcosh-cosx-sinxsinh)/h##

## = lim_(hrarr0)((cosxcosh-cosx)/h-(sinxsinh)/h)##

## = lim_(hrarr0)(cosx(cosh-1)/h-sinx(sinh)/h)##

## = cosx(lim_(hrarr0)(cosh-1)/h)-sinx(lim_(hrarr0)(sinh)/h)##

## = cosx(0)-sinx(1)##

## = -sinx##