# The algebra of differentiable functions.

Theorem 6.5.1   If and are two differentiable functions at , then is differentiable at and

For example, take and as follows:

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 ).

Theorem 6.5.2   If and are two differentiable functions at , then is differentiable at and

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.

Example 6.5.3   Let . Then we have:

For example, take and as follows:

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.

Theorem 6.5.4   If ia a differentiable function at , and if , then is differentiable at and

Example 6.5.5   Let and . The cosine function is differentiable at 0 and . Thus, is differentiable at 0 and .

Theorem 6.5.6   If and are two differentiable functions at , and if , then is differentiable at and

Example 6.5.7   Let . The we have:

Example 6.5.8   Let . As the cosine and sine function are differentiable at 0 and , the tangent function is differentiable at 0; we have:

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

Theorem 6.5.9   If is differentiable at and if is differentiable at , then is differentiable at , and

Example 6.5.10   Let . According to the notations of Theorem 5.9, is a polynomial function and is the sine function. Both are differentiable over the whole of . Let us differentiate the function ;

For example, take the function defined as follows:

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

Remark 6.5.11 (Chain Rule)   We will use the notations of 1.4 with the following setting:

Then:

Example 6.5.12   Let and . Then for every , we have:

Example 6.5.13   Let and . Then for every , , hence the function is defined by and is differentiable at every point in . We have:

This last example is an illustration of the following result:

Corollary 6.5.14   Let be a function, differentiable on the domain . Suppose that for any . Then the function is differentiable on and we have:

Example 6.5.15   Let . As the radicand is positive for any real number , the function is defined over and is differentiable over . Now let us differentiate the function:

Another very useful result is the following:

Corollary 6.5.16   Let be a function, differentiable on the domain . Then is differentiable on and we have:

Example 6.5.17   Let . This function is defined, and differentiable, on , as the polynomial has only positive values.

We have:

Of course we can iterate the process.

Example 6.5.18   Let . The function is defined whenever and and , i.e. . As we compose three differentiable functions, the function is differentiable over , and we have:

Example 6.5.19   Let . This function is defined, and differentiable, on .

We have:

Example 6.5.20   Let . This function is defined, and differentiable on (please, check this!).

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:

Corollary 6.5.21   Let be a function defined over the interval I; denote . Suppose that the following properties are verified:
(i)
is invertible;
(ii)
is differentiable at ;
(iii)
.
If , then is differentiable at and

Example 6.5.22   The function such that for is invertible and .

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 :

Example 6.5.23 (Inverse trigonometric functions)

(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