A Modified Power Rule (The Chain Rule):     

• If , then after first multiplying
 

 

We can then find the derivative by

 

 

• There is another approach to this derivative which helps us with more complicated functions.
   
  If we think of  as made up of two functions, and  , we could try to follow the power rule for the square.
   
We get
 
But this isn’t right since we already know the correct answer.
 
But if we start with  and multiply by the derivative of  which is  we have:
 

 
which is a factored version of our correct answer
 

 
• Whenever and  is another function of  we can use this new method.

The Chain Rule

Let's Practice:
 
1. If .

One advantage of using this formula is that we get an already (at least partially) factored version of the derivative.

2.  In order to get the derivative, we first rewrite this as

 
Now we apply the chain rule and we get
 

Calculate the derivatives of each function. Write in fraction form, if needed, so that all exponents are positive in your final answer. Use the "modified power rule" for each.