Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 14



Differentiability

In this section, we consider functions

where is a subset of the real numbers. We want to explain what it means that such a function is differentiable in a point . The intuitive idea is to look at another point , and to consider the secant, given by the two points and , and then to let " move towards “. If this limiting process makes sense, the secants tend to become a tangent. However, this process only has a precise basis, if we use the concept of the limit of a function as defined earlier.


Definition  

Let be a subset, a point, and

a function. For , , the number

is called the difference quotient of for

and .

The difference quotient is the slope of the secant at the graph, running through the two points and . For , this quotient is not defined. However, a useful limit might exist for . This limit represents, in the case of existence, the slope of the tangent for in the point .


Definition  

Let be a subset, a point, and

a function. We say that is differentiable in if the limit

exists. In the case of existence, this limit is called the derivative of in , written

The derivative in a point is, if it exists, an element in . Quite often one takes the difference as the parameter for this limiting process, that is, one considers

The condition translates then to , . If the Function describes a one-dimensional movement, meaning a time-dependent process on the real line, then the difference quotient is the average velocity between the (time) points and and is the instantaneous velocity in .


Example

Let , and let

be an affine-linear function. To determine the derivative in a point , we consider the difference quotient

This is constant and equals , so that the limit of the difference quotient as tends to exists and equals as well. Hence, the derivative exists in every point and is just . The slope of the affine-linear function is also its derivative.


Example

We consider the function

The difference quotient for and is

The limit of this, as tends to , is . The derivative of in is therefore .



Linear approximation

We discuss a property which is equivalent with differentiability, the existence of a linear approximation. This formulation is important in many respects: It allows giving quite simple proofs of the rules for differentiable functions, one can use it to reduce differentiability to the continuity of an error function, it yields a model for approximation with polynomials of higher degree (quadratic approximation, Taylor expansion), and it allows a direct generalization to the higher-dimensional situation(in the second term)


Theorem

Let be a subset, a point, and

a function. Then is differentiable in if and only if there exists some and a function

such that is continuous in , , and such that

Proof  

If is differentiable, then we set

Then the only possibility to fulfill the conditions for is

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Because of differentiability, the limit

exists, and its value is . This means that is continuous in .
If and exist with the described properties, then for the relation

holds. Since is continuous in , the limit on the left-hand side, for , exists.


The affine-linear function

is called the affine-linear approximation. The constant function given by the value can be considered as the constant approximation.


Corollary

Let be a subset, a point, and

a function. Then is also continuous in .

Proof  

This follows immediately from Theorem 14.5 .



Rules for differentiable functions
An illustration of the product rule: the increment of the area is about the seize of the sum of the two products of the side length and the increment of the other side length. For the infinitesimal increment, the product of the two increments is irrelevant.

Lemma

Let be a subset, a point, and

functions which are differentiable

in . Then the following rules for differentiability holds.
  1. The sum is differentiable in , with
  2. The product is differentiable in , with
  3. For , also is differentiable in , with
  4. If has no zero in , then is differentiable in , with
  5. If has no zero in , then is differentiable in , with

Proof  

(1). We write and respectively with the objects which were formulated in Theorem 14.5 , that is

and

Summing up yields

Here, the sum is again continuous in , with value .
(2). We start again with

and

and multiply both equations. This yields

Due to Lemma 10.11 for limits, the expression consisting of the last six summands is a continuous function, with value for .
(3) follows from (2), since a constant function is differentiable with derivative .
(4). We have

Since is continuous in , due to Corollary 14.6 , the left-hand factor converges for to , and because of the differentiability of in , the right-hand factor converges to .
(5) follows from (2) and (4).


These rules are called sum rule, product rule, quotient rule. The following statement is called chain rule.


Theorem

Let denote subsets, and let

and

be functions with . Suppose that is differentiable in and that is differentiable in . Then also the composition

is differentiable in , and its derivative is

Proof  

Due to Theorem 14.5 , one can write

and

Therefore,

The remainder function

is continuous in with value .


An illustration for the derivative of the inverse function. The graph of the inverse function is the reflection of the graph at the diagonal, and the tangent behaves accordingly.

Theorem

Let denote intervals, and let

be a bijective continuous function, with the inverse function

Suppose that is

differentiable in with . Then also the inverse function is differentiable in , and

holds.

Proof  

We consider the difference quotient

and have to show that the limit for exists, and obtains the value claimed. For this, let denote a sequence in , converging to . Because of Theorem 11.7 , the function is continuous. Therefore, also the sequence with the members converges to . Because of bijectivity, for all . Thus

where the right-hand side exists, due to the condition, and the second equation follows from Lemma 8.1 .



Example

The function

is the inverse function of the function , given by (restricted to ). The derivative of in a point is . Due to Theorem 14.9 , for , the relation

holds. In the zero point, however, is not differentiable.


Example

The function

is the inverse function of the function , given by . The derivative of in is , which is different from for . Due to Theorem 14.9 , we have for the relation

In the zero point, however, is not differentiable.



The derivative function

So far, we have considered differentiability of a function in just one point, now we consider the derivative in general.


Definition  

Let denote an interval, and let

be a function. We say that is differentiable, if for every point , the derivative of in exists. In this case, the mapping

is called the derivative of .


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