The functions and are not differentiable at 0 (as they are not continous at 0, see theorem 7.1), but is differentiable at 0 ( is constant on ).

If and are two functions differentiable on the same domain , then is differentiable on and we have:

This is the so-called Leibniz's formula.

The functions and are not differentiable at 0, but is differentiable at 0 ( is constant on ).

Using the fact that a constant function is differentiable on its domain, Theorems 5.1 and 5.2 imply that the set of all the differentiable functions on the interval is a real vector space.

If and are two functions differentiable on the same domain , then is differentiable on and we have:

At the point 2, is not differentiable. Now take a function constant over , say . Then is constant over , thus is differentiable over .

Another very useful result is the following:

We have:

Of course we can iterate the process.

We have:

We have:

An important corollary of Theorem 5.9 is the following; it enables us to differentiate the inverse function (under certain conditions) of a known function:

- (i)
- is invertible;
- (ii)
- is differentiable at ;
- (iii)
- .

As , Corollary 5.21 does not apply for at ). At every other (positive) point, the corollary applies, and we have:

We obtained already this derivative by applying Definition 1.1.

For example, Figure 4 displays the tangents at to the graphs of and of :

- (i)
- Let
, for
. As the derivative of the sine function is equal to 0 at the points
and
, we cannot apply
Corollary 5.21 for
at
and at
.
At any other point in the open interval , Corollary 5.21 applies, and we have:

by example identities with inverse trigonometric functions

- (ii)
- Let
, for
. By an argument similar to the previous one, we conclude that this function is differentiable over
and that

- (iii)
- Let
, for
. The tangent function is differentiable over the interval
and its first derivative never vanishes. Therefore Corollary 5.21 applies at any real point and we have:
by example identities with inverse trigonometric functions

The formulas that we showed here should be added to the table of derivatives of elementary functions.

Noah Dana-Picard 2007-12-28