Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 16



Derivative of power series

Many important functions, like the exponential function or the trigonometric functions, are represented by a power series. The following theorem shows that these functions are differentiable, and that the derivative of a power series is itself a power series, given by differentiating the individual terms of the series.


Theorem

Let

denote a power series which converges on the open interval , and represents there a function . Then the formally differentiated power series

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is convergent on . The function is differentiable in every point of the interval, and

holds.

Proof

The proof requires a detailed study of power series.


In the formulation of the theorem, we have distinguished between for the power series and for the function, defined by the series, in order to stress the roles they play. This distinction is now not necessary anymore.


Corollary

A function given by a power series is infinitely often differentiable on its interval of convergence.

Proof  

This follows immediately from Theorem 16.1 .



Theorem

The exponential function

is

differentiable with

Proof  

Due to Theorem 16.1 , we have

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Theorem

The exponential function

with base , is differentiable with

Proof  

By definition, we have

The derivative with respect to equals

due to Theorem 16.3 and the chain rule



Remark

For a real exponential function

the relation

holds, due to Theorem 16.4 . Hence, there is a proportional relationship between the function and its derivative , and is the factor. This is still true if is multiplied with a constant. If we consider as a function depending on time , then describes the growing behavior at that point of time. The equation means that the instantaneous growing rate is always proportional with the magnitude of the function. Such an increasing behavior (or decreasing behavior, if ) occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable (the number of mice is then proportional with the number of mice born). A condition of the form

is an example of a differential equation. This is an equation for a function, which expresses a condition for the derivative. A solution for such a differential equation is a differentiable function which fulfills the condition on its derivative. The differential equation just mentioned are fulfilled by the functions

We will study differential equations in the second semester.


Corollary

The derivative of the natural logarithm

is

Proof  

As the logarithm is the inverse function of the exponential function, we can apply Theorem 14.9 and get

using Theorem 16.3 .



Corollary

Let . Then the function

is differentiable, and its derivative is

Proof  

By definition, we have

The derivative with respect to equals

using Theorem 16.3 , Corollary 16.6 and the chain rule



Theorem

The sine function

is differentiable, with

and the cosine function

is differentiable, with

Proof



Theorem

The tangent function

is differentiable, with

and the cotangent function

is differentiable, with

Proof  

Using the quotient rule Theorem 16.8 , and the circle equation we get

The derivative of the cotangent function follows in the same way.



The number


The number is the area and half of the circumference of a circle with radius . But, in order to build a precise definition for this number on this, we would have first to establish measure theory or the theory of the length of curves. Also, the trigonometric functions have an intuitive interpretation at the unit circle, but also this requires the concept of the arc length. An alternative approach is to define the functions sine and cosine by their power series, and then to define the number with the help of them, and establishing finally the relation with the circle.


Lemma

The cosine function has, within the real interval , exactly one zero.

Proof  

We consider the cosine series

For , we have . For , one can write

Hence, due to the intermediate value theorem there exists at least one zero in the given interval.
To prove uniqueness, we consider the derivative of cosine, which is

due to Theorem 16.8 . Hence, it is enough to show that sine is positive in the interval , because then cosine is strictly decreasing by Theorem 15.7 in the interval and there is only one zero. Now, for , we have



A rational approximation of the number on a -pie.

Definition  


Let denote the unique (according to Lemma 16.10 ) real zero of the cosine function in the interval . Then the number is defined by


Theorem

The sine function and the cosine function fulfill in the following periodicity properties.
  1. We have and for all .
  2. We have and for all .
  3. We have and for all .
  4. We have , , , , and .
  5. We have , , , , and .

Proof  

Due to the circle equation

we have , hence , because of the reasoning in the proof of Lemma 16.10 . From that we deduce, with the help of the addition theorems the relations between sine and cosine as mentioned in (3), e.g.

Hence it is enough to prove the statements for cosine. All statements follow from the definition of and from (3).



Definition  

A function is called periodic with period length , if the equality

holds for all

.

The trigonometric functions sin and cosine are periodic functions with the period length .



The inverse trigonometric functions

Corollary

The real sine function induces a bijective, strictly increasing function

and the real cosine function induces a bijective, strictly decreasing function

Proof



Corollary

The real tangent function induces a bijective, strictly increasing function

and the real cotangent function induces a bijective strictly decreasing function

Proof


Due to the bijectivity of sine, cosine, tangent and cotangent on suitable interval, there exist the following inverse functions.


Definition  

The inverse function of the real sine function is

and is called arcsine.

Definition  

The inverse function of the real cosine function is

and is called arccosine.
Arkustangens
Arkustangens

Definition  

The inverse function of the real tangent function is

and is called arctangent.

Definition  

The inverse function of the real cotangent function is

and is called arccotangent.

Theorem

The inverse trigonometric functions have the following

derivatives.

Proof  

For example, for the arctangent, we have, due to Theorem 14.9 ,


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