Quoted By:
df(x)/dx = lim_{h->0} [f(x+h)-f(x)]/h
dsin(x)/dx = lim_{h->0} (sin(x+h)-sin(x))/h
= lim_{h->0} [sin(x)cos(h) + sin(h) cos(x)-sin(x)]/h
= lim{h->0} sin(x) [cos(h)-1]/h + sin(h)/h cosx
= cosx
The derivative f'(x) is not tangent to the function f(x) when plotted - f'(x) gives the gradient of the tangent line of the function f(x) at the point x. Ie plot cos(x), at x=0 cos(x) is clearly flat. sin(0) = 0, ie the tangent line of cos(x) at x=0 has gradient 0 - a flat line.
