Introductory Calculus: The Chain Rule
By M Ransom
Note that in this lesson we show examples, but not any proof that the formulas for the product and quotient rules are correct.
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.- Whenever
and 
is another function of 
we can use this new method.
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:
and multiply by the derivative of which is we have:
which is a factored version of our correct answer
- If
.
Let's Practice:
One advantage of using this formula is that we get an already (at least partially) factored version of the derivative.
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.
