# 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.

Let be a subset, a point, and

a function. For , , the number

is called the *difference quotient* of for

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 .

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.

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)

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

If is differentiable, then we set

Then the only possibility to fulfill the conditions for is

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.

This follows immediately from Theorem 14.5 .

*Rules for differentiable functions*

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

(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*.

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

Due to Theorem 14.5 , one can write

and

Therefore,

The remainder function

is continuous in with value .

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

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 .

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.

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.

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

*derivative*of .

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