Chain Rule

The chain rule is one of the most commonly used rules in differential calculus. Essentially it is used to find the derivative of composite functions. So what does that mean exactly? Well let’s take a look.

Suppose we have the function f(x) = sqrt(x^2 + 9), how would we go about finding its derivative? The calculus tools you have learned up until this point do not allow you to solve this. So as you may have guessed we have to use the chain rule.

The chain rule makes use of inherent substitution to solve composite functions. We can write this out in Leibniz form as;

dy/dx = (dy/du) * (du/dx)

This may look kind of complicated, but it is pretty easy to put into use.

Let’s go back to our function f(x) = sqrt(x^2 + 9). First we can make the substitution u = x^2+9. We will refer to this as our inside function. Now if we sub u back into our original function we are left with;

f(u) = sqrt(u)

So according to our chain rule we need to find two terms; dy/du and du/dx. Dy/du is the derivative of the ‘outside’ function. We can solve this fairly easily.

Dy/du = 1/(2*sqrt(u)). This is a pretty common derivative, hopefully you have seen this before.

So now we can find the derivative of the inside function;

u = x^2 + 9
Du/dx = 2x

Now that we have both pieces, we can put them into the chain rule.

Dy/dx = 1/(2sqrt(u)) * 2x
= 2x/(2sqrt(x^2+9))
= x /sqrt(x^2 + 9)

So that’s really all there is to it. Hopefully now you can see that this is easier than it sounds. Find some more example problems and practice it more. It is an important rule you will have to use often.