Section 3.6: The Chain Rule

Abstract:

How do you differentiate a composition? The Chain Rule holds the key.

The chain rule

Suppose that we have dependent variable y defined as a function of independent variable x

but that x is itself a variable of t: x=g(t). Then we can actually think of y as a function of t, as a composition

If we want to calculate the derivative , we can proceed in different ways:

1. We could find the formula g in x=g(t), then substitute for the variable x in the formula f(x). Example:

   

   You see that x has been eliminated from the picture. But now we have a composition - what next?

2. Alternatively, we could reason as follows:

   

   (one of my favorite tricks - multiply by an appropriate form of 1), so that

   

   Now, if

   

   then

   

   In the example from above,

   

   and if we want to express this in terms of t, we can substitute back in for x as a function of t at the end:

   

Another notation

Another quite popular notation for the chain rule comes right out of Leibniz's notation - i.e.,

If y=f(x) and x=g(t), then

Notice that the primes on the functions above refer to derivatives with respect to different variables - a potential source of confusion. The prime on f refers to the derivative with respect to x, whereas the prime on g refers to the derivative with respect to t.