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 .
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
- Failed to parse (unknown function "\begin{cases}"): {\displaystyle {{}} r(x) = \begin{cases} \frac{ f (x )-f (a) }{ x -a } - s \text{ for } x \neq a\, , \\ 0 \text{ for } x <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 14]] __NOINDEX__ a \, . \end{cases} \, }
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.
- Rules for differentiable functions
Lemma
Let be a subset, a point, and
functions which are differentiable
in . Then the following rules for differentiability holds.- The sum is differentiable in , with
- The product is differentiable in , with
- For
,
also is differentiable in , with
- If has no zero in , then is differentiable in , with
- 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 .
Theorem
Let denote intervals, and let
be a bijective continuous function, with the inverse function
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
<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this lecture Exercise sheet for this lecture (PDF) |
---|