Thenand is differentiable at a point and both have all derivatives of orders up to in a neighborhood of . Then, is differentiable at with
     
Proof:

It is easy to prove the required statement by induction on . We leave the details to the reader.

Let us use Leibnitz's rule to find the thired derivative of the function

Let
             
Then

            
and
         .

Thus

We saw that the derivative of a function also represents the rate of change of the function. This interpretation along with the chain rule is useful in solving problems which involve various rates of change.