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##
