If f(x)=sin(x), then f(x+h)=sin(x+h)=sin(x)cos(h)+cos(x)sin(h).
f(x+h)-f(x)=sin(x)cos(h)+cos(x)sin(h)-sin(x)=
sin(x)(cos(h)-1)+cos(x)sin(h).
Divide both sides by h:
(f(x+h)-f(x))/h=sin(x)(cos(h)-1)/h+cos(x)sin(h)/h.
[This is usually followed by finding the limit of (f(x+h)-f(x))/h as h→0. cos(h)→1-h2/2 as h→0, so (cos(h)-1)/h→(-h2/2)/h=-h/2→0 as h→0.
sin(h)→h as h→0, so sin(h)/h→h/h=1.
Therefore, sin(x)(cos(h)-1)/h+cos(x)sin(h)/h→0+cos(x)=cos(x), which is in fact the derivative of sin(x).]