What is the "chain rule"?

The "chain rule" is a handy little tool that we can use to find the derivative of a complicated function.  Specifically, we use the chain rule when we have functions within functions.  But what does that mean?

All functions start with a variable, lets call that variable x.  Now, a function is just a set of operations performed on x which changes it in some way. For example   " 2x +1 " ,  "x^3" and  " Sin(x) "  are all functions of x.   These functions are relatively easy to take the derivative of (if you don't know how then get in touch with me and we'll have you doing them in no time!).  But how do we take the derivative of a function like " Sin(x^3) " , this looks complicated right?

If we look closely we can see that it's actually a function of x (" x^3" ) INSIDE another function (the Sin() function).  This is where we use the CHAIN RULE.  It makes our life much easier because it allows us to take the derivative of complicated functions like " Sin(x^3) " by breaking it down into smaller chunks which we know how to differentiate!

This means that if we know how to differentiate x^3 with respect to x, and we know how to differentiate Sin(x) with respect to x, then we actually know how to differentiate Sin(x^3)  with respect to x as well.

I can explain this in more detail in a free trial session on the mytutor website,  so please get in touch if you have any questions!