Our starting point is the following limit:
To see why, it is enough to rewrite the expression involving the cosine as
But , so we have
This limit equals and thus .
In fact, we may use these limits to find the derivative of and at any point x=a. Indeed, using the addition formula for the sine function, we have
So we have proved that exists and .Similarly, we obtain that exists and that .
Since , , , and are all quotients of the functions and , we can compute their derivatives with the help of the quotient rule:
It is quite interesting to see the close relationship between and (and also between and ).
From the above results we get
These two results are very useful in solving some differential equations.Example 1. Let . Using the double angle formula for the sine function, we can rewrite
So using the product rule, we get
which implies, using trigonometric identities,
In fact next we will discuss a formula which gives the above conclusion in an easier way.