Chain Rule:

If y is a function of u and u is a function of x, then

                               

 

Another important result:

                    

 

This rule is very important in differentiation. It is used in implicit differentiation, finding rates of changes, as well as parametric differentiation.

 

Here we introduce the chain rule, but it is hard to give examples without going into its various applications, which have yet to be explained. Hence, detailed explanations will be provided in the relevant sections.

 

We have listed an example of the chain rule in this section so far, in the first section, Differentiation. We have stated that, given the equation y = uⁿ, the derivative is

                             

Since dy/du = nu^n-1, it can be seen that this relation uses the chain rule.