The Chain Rule

As it's name indicates, a chain of functions in Math can be thought of as Compositions of functions.  It is precisely for compositions of functions that we use the Chain Rule.  The Chain Rule is a rule in elementary Calculus that allows us to find the derivative of a function that is a composition of two functions.

The Chain Rule says:

If h(x)=f(g(x))  That is, h(x)= f o g (x)  or in another way, h(x)= f(x) o g(x) All the same symbols and meaning, h(x) is the composition of f and g (x), then:

h'(x)=f'(g(x))*g'(x)

As a way to remember this, when can think that "when we want the derivative of h(x), the composite of f(x) and g(x), then, we do the derivative of the outside function keeping the inside function untouched, then, we do, times the derivative of the inside function."

Note here that by the outside function, we are refering to f(x) and the inside function g(x).


Now for an examle:  Take h(x)=sin(x^2)  So, here the two functions are f(x)=sin(x) and g(x)=x^2.  And we have f o g(x)=h(x).

so,

h'(x)=cos(x^2)*2x

Notice here that the derivative of the outside function keeping the inside function untouched, that is, the derivative of sin(x^2) is cos(x^2) and then, we go to the inside function, which is g(x)=x^2, and the derivative of x^2 is 2x, which as the chain rule describes is the derivative of the outside function.