Real Analysis – Differential Calculus II
Theorem 60 (Differentiability of the composite function) Let , , and . If is differentiable in and is differentiable in , then in and it is
Using Leibniz’s notation we can also write the previous theorem as A notation that formally suggests that we can cancel out the . Proof: Let . Since is differentiable in by Theorem 57 it is with continuous in . Taking and it is Hence Since is differentiable in it also is continuous in by Corollary 59. Then also is continuous in (by Theorem 43). Hence Taking the limit in 27 it is Which is to say

As an application of Theorem 60 let us look into some simple examples.

Now and let . Hence
Hence
 Let and and calculate .
which generalizes the know rule for integer exponents.
Hence

Like in the first example the construction of interest is where and .
Hence
Hence for
In particular one can calculate
Since for it always is
Just like in Theorem 60 we will state an application of the previous theorem.
Let and , then .
Now
 is differentiable in all points of the interval
 for all points contained in the interval
 is continuous
Then
Finally
Or, writing in a notation that is more usual
In general one can define superior derivatives by using recursion.
Let us denote the nth derivative of by (since for the 50th derivative it isn’t very practical to use fifty times). One first define . Now for it is
That is to say that
 …
Given the previous discussion it makes sense to introduce the following definition
Definition 39 A function is said to be times differentiable in if all exist and are finite. 
We already know that a differentiable function is continuous (via Corollary 59 in Real Analysis – Differential Calculus I) , but is it that the derivative of a differentiable function also is a continuous function?
As an (counter)example let us look into the following function:
It is easy to see that is differentiable in
but isn’t continuous for . The reader is invited to calculate and .
Apparently the derivative of a function either is continuous or it is strongly discontinuous. That being said it is obvious that it makes sense to introduce differentiability classes, which classifies a function according to its derivatives properties.
Definition 40 A function is said to be of class if it is times differentiable and is continuous. 
It is easy to see that a function that is of class also is of classs .
A function is said to be of class if it has finite derivatives in all orders (which are necessarily continuous).
If are times differentible in then , , are also times differentiable in .
Definition 41 Let , and . is said to be a relative maximum of if 
Definition 42 Let , and . is said to be a relative minimum of if 
Theorem 62 (Interior Extremum Theorem) Let and is an interior point of . If has a relative extremum in and exists then
Proof: Let us suppose without loss of generality that has a relative maximum in . Since is an interior point of and exists, and exist and are equal. It is From our hypothesis Hence Then by corollary 31 (Real Analysis – Limits and Continuity II) it is Likewise Hence Since it has to be and consequently . 
One can visualize the previous theorem in the following way. Imagine that you have a relative maximum in a given interval for a continuous function. For some vicinity of that point we must have function values that are inferior to . Since we are assuming that is a maximum values to its left are increasing as we approach and values to its right are decreasing as we mover further away from .
Hence for its left side the derivative of has positive values, while to its left the derivative of has negative values, since we also assume that the derivative exists in we can reason by continuity that its value is .
Theorem 63 (Rolle’s Theorem) Let and continuous such as . If is differentiable and there exists a point such as .
Proof: Since is continuous in the compact interval it has a maximum and a minimum in (see Extreme Value Theorem which is theorem 55 in Real Analysis – Limits and Continuity VII). If for is either a maximum or a minimum then by Theorem 62 . Let denote the minimum and denote the maximum. and let us analyze the case were the extrema values occur at the extremities of the interval. Since by hypothesis then . In this case is constant and it trivially is 
Corollary 64 Let , continuous such as . If is differentiable in the interior of and doesn’t vanish in the interior of , then doesn’t have more than one in .
Proof: Let us use the method of reductio ad absurdum that vanishes for two points of ( and ). applying Theorem 63 in (since ) there exists in such as . Hence vanishes in the interior of which contradicts our hypothesis. 
Leave a Reply