Proof of f(g(x)) = f(g) * g(x) from the definition

We can use the definition of the derivative:

f(x) =

lim d-->0

f(x+d)-f(x) d

f(g(x)) = df/dx = (f(g(x+d) - f(g(x))/d

df/dx * 1/(dg/dx) = [ (f(g(x+d) - f(g(x))/d ] * [ d/(g(x+d) - g(x)) ]
= ( f(g(x+d))-f(g(x)) )/(g(x+d)-g(x)) = df/dg

df/dx = df/dg * dg/dx