Math2.org Math Tables: Chain Rule

(Math)
(d/dx) f(g(x)) = (d/dg) f(g) * (d/dx) g(x)


Proof of (d/dx) f(g(x)) = (d/dg) f(g) * (d/dx) g(x) from the definition

We can use the definition of the derivative:

(d/dx) f(x) = lim
d-->0  		 f(x+d)-f(x)
d
Therefore, (d/dx) f(g(x)) can be written as such:
(d/dx) 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