The derivative is one of the central concepts within calculus. For a given function , the derivative is another function which specifies the rate of change of in . It is used in various scientific disciplines, basically everywhere, where there is a "rate of change" within a dynamical system. Knowing about derivatives means having a powerful tool at hand: it allows you to describe and predict rates of change in a huge variety of applications.

Intuitions of the derivative Bearbeiten

Te derivative is a mathematical object, which becomes useful in many situations. Depending on the situation, there are several intuitions which can make this abstract object come alive in your mind:

  • Derivative as instantaneous rate of change: The derivative corresponds to what we intuitively understand as the rate of change of a function   at some instant  . A rate of change ( ) describes how much a quantity changes ( ) in relation to the change of some reference quantity ( ). If we let ( ) run to 0, we get the rate of change within an "infinitely small amount of time". An example are speeds: Consider a given time-dependent position  , i.e. the function   is re-labeld as   and   is re-labelled as  . The quotient   of "travelled distance"   and "elapsed time"   just describes the "average speed". In order to get the speed   at some time  , we make the time difference   smaller and smaller, such that the "average speed"   goes over to an "instantaneous speed"   . This   is called first derivative and mathematicians write   .
  • Derivative as tangent slope: The derivative corresponds to the slope that the tangent of the graph has at the location of the derivative. Thus the derivative solves the geometric problem of determining the tangent to a graph by a point.
  • Derivative as slope of the locally best linear approximation: Any function that has a derivative a point can be well approximated by a linear function in an environment around this point. The derivative corresponds to the slope of this linear function. This is useful if the function is hard to compute: the linear approximation can be computed way easier in many cases.
  • Derivative as generalised slope: How steep is a given function? At first, the concept of the "slope of a function" is only defined for linear functions. But we can use the derivative to define the "slope" also for non-linear functions.

We will discuss these intuitions in detail in the following and use them to derive a formal definition of the derivative. We will also see that derivable functions are "kink-free", which is why they are also called smooth functions (think of smoothly bending some dough or tissue).

Derivative as rate of change Bearbeiten

Introduction to the derivative Bearbeiten

The derivative corresponds to the rate of change of a function  . How can this rate of change of a function be determined or defined? Let, for example be   a real-valued function, which has the following graph:

 
The function f

For example,   may describe a physical quantity in relation to another quantity. For example,   could correspond to the distance covered by an object at the time  .   could also be the air pressure at the altitude   or the population size of a species at the time  . Now let us take the argument  , where the function has the function value  :

 
The function f with one argument and function value

Let us assume that   is the distance travelled by a car at the time  . Then the current rate of change of   at the position   is equal to the velocity of the car at the time  .

It is hard to determine the velocity directly with only   given. But we can estimate it. We take a point in time   shortly after   and look at the average speed in that time  . The distance travelled in that time is  , while the time difference is  . Thus the car has the average speed

 

This quotient, which indicates the average rate of change of the function   in the interval  , is called difference quotient. As its name suggests, it is a quotient of two differences. In the following figure we see that this difference quotient is equal to the slope of the secant passing through the points   and  :

 
The average rate of change corresponds to the slope of the secant.

This average speed is a good approximation of the current speed of our car at the time  . It is only an approximation since the movement of the car between   and   need not be uniform - it can accelerate or decelerate. But we should get a better result if we shorten the period for calculating the average speed. So let's look at a time   which is even closer to   and determine the average speed   for the new time interval between   and  :

 
The secant for a point closer to the derivative argument

We can shorten the time difference even further by taking a sequence   of times which converge towards  . For every   we calculate the average speed   of the car in the period from   to  . The shorter  , the less the car should be able to accelerate or decelerate in this period of time. So the average speed should converge to the current speed of the car at time  :

 
The secant slope (average rate of change) converges to the derivative (current rate of change).
 
For   , the average rate of change   converges to the current rate of change  .

Thus we have found a method to determine the current rate of change of   at time  : We take any sequence of arguments  , which are all different from   and for which  . For every   we determine the quotient  . The current rate of change is the limit of these quotients:

 

The derivative or   at   is denoted as  . So we have the mathematical definition:

 

The limit of the difference quotient is sometimes also called differential quotient.

Negative time intervals Bearbeiten

What happens if we do not choose   in the future, but in the past of  ? Let us draw this situation in a picture:

 
The average rate of change for an argument lower than the derivative argument

The average speed in the interval from   to   is then equal to  . If we extend this fraction by a factor of  , we get

 

We get the same term as in the previous section. This gives the average speed, no matter if   or  . Thus, in the case of a negative time interval with   the average speed should also be close to the current speed of the car at the time  , if   is only sufficiently close to  . There is

 

where   is any sequence of different from   with  . The sequence elements of   can sometimes be larger and sometimes smaller than   depending on the index  :

 
A sequence of secants for computing the derivative

Refining the definition Bearbeiten

Let now   be a real-valued function and let  . As we have seen above, there is

 

where   is a sequence of arguments different from   which converges to  . In order to have at least one such sequence of arguments,   must be an accumulation point of the domain   (an element is an accumulation point of a set exactly when there is a sequence not including that number but converging towards it). This may sound more complicated than it often is. In most cases   is an interval and then every   is an accumulation point of  . For the definition of the differential quotient it should not matter which sequence   we choose. Accordingly, we can define the derivative:

Let   with   and let   be an accumulation point of  . The function   is differentiable at   with derivative  , if for every sequence   of arguments different from   and with   there is:

 

We can shorten this definition by using limits for functions. As a reminder: There is according to definition:   if and only if   for all sequences   of arguments non-equal to   with  . So:

Let   with   and let   be an accumulation point of  . The function   is differentiable at   with derivative  , whenever:

 

The h-method Bearbeiten

 
Definition of the derivative via the h-method: For the respective h-values the corresponding secants are drawn in. You see that for   the secant changes into the tangent and thus the secant slope (difference quotients) changes into the tangent slope (derivative).

There is an equivalent option to define the derivative. For this we go from the differential quotient   and perform the substitution  . The new variable   just describes the difference between   and the point where the difference quotient is formed. For  , equivalently goes  . So we can also define the derivative as follows

Let   with   and let   be an accumulation point of  . The function   is differentiable at   with derivative  , whenever:

 

Applications in science and technology Bearbeiten

We have come to know the derivative as the current rate of change of a quantity. As such, it occurs frequently in science or applications. Several variables are defined as rates of change, for example:

  • velocity: The velocity is the instantaneous rate of change of the distance travelled by an object.
  • Acceleration: The acceleration is the instantaneous rate of change of the speed of an object.
  • Pressure change: Let   the air pressure at altitude  . The derivative   is the rate of change of air pressure with altitude. This example shows that the rate of change need not always be related to time. It can also be the rate of change with respect to another quantity, e.g. altitude.
  • Chemical reaction rate: Let's consider a chemical reaction  . Let   the concentration of the substance   at time  . The derivative   is the instantaneous rate of change of the concentration of   and thus indicates how much of the substance   is converted into the substance  . Thus   indicates the chemical reaction rate for the reaction  .
  • Often the number of individuals   in a population is considered (for example the number of people on the planet, the number of bacteria in a Petri dish, the number of animals of a species or the number of atoms of a radioactive substance). The derivative   represents the instantaneous rate of change of individuals at the time  .

Definitions Bearbeiten

Derivative and differentiability Bearbeiten

Definition

Let   with   and let   be an accumulation point of  . The function   is differentiable at   with derivative  , whenever:

 

Equivalently, we can require:

 

A function that can be differentiated at   is called differentiable at the position  . A function is called differentiable, if the above limit exists at every position within the domain of definition. That means, differentiable functions are differentiable at every point, where they are defined.

Difference quotient and differential quotient Bearbeiten

 
Der difference quotient between   and   is just the slope of the blue secant

The terms „difference quotient“ and „differential quotient“ are mathematically defined as follows:

 
The following definitions hold:

Definition (Difference quotient)

The difference quotient of a function   with   for the interval   is the quotient

 

This quotient corresponds to the slope of the secant between the points   and  .

Definition (Differential quotient)

Let   with  . Let   be an accumulation point of the domain  . The differential quotient of this function at   is defined as the limit:

 

If the limit exists, it coincides with the derivative  .

Derivative function Bearbeiten

 
The derivative function assigns to every argument   of the function   its derivative  . Within the animation, the derivative function is evaluated at several arguments. It corresponds to the slope of the tangent at those points.

If a function   with   is differentiable at every point within its domain of definition, then   has a derivative at every point in  . The function that assigns its derivative   to every m argument   is called derivative function of  :

Definition (Derivative function)

Let   be a differentiable function with  . We define the derivative function   by

 

If the derivative function   is additionally continuous, we call   continuously differentiable.

Warning

The terms "continuously differentiable" and "differentiable" are not equivalent. The continuity of the derivative function has to be imposed separately.

Notations Bearbeiten

Historically, different notations have been developed to represent the derivative of a function. In this article we have only learned about the notation   for the derivative of  . It goes back to the mathematician Joseph-Louis Lagrange , who introduced it in 1797. Within this notation the second derivative of   is denoted   and the  -th derivative is denoted   .

Isaac Newton - (the founder of differential calculus besides Leibniz) - denoted the first derivative of   with  , accordingly he denoted the second derivative by  . Nowadays this notation is mainly used in physics for the derivative with respect to time.

Gottfried Wilhelm Leibniz introduced for the first derivative of   with respect to the variable   the notation  . This notation is read as "d f over d x of x". The second derivative is then denoted   and the  -th derivative is written as  .

The notation of Leibniz is mathematically speaking not a fraction! The symbols   and   are called differentials, but in modern calculus (apart from the theory of so-called "differential forms") they have only a symbolic meaning. They are only allowed in this notation as formal differential quotients. Now there are applications of derivatives (like the "chain rule" or "integration by substitution"), in which the differentials   or   can be handled as if they were ordinary variables and in which one can come to correct solutions. But since there are no differentials in modern calculus, such calculations are not mathematically correct.

The notation   or   for the first derivative of   dates back to Leonhard Euler. In this notation, the second derivative is written as   or   and the  -th derivative as   or  .

Overview about notations Bearbeiten

Notation of the … 1st derivative 2nd derivative  -th derivative
Lagrange      
Newton      
Leibniz      
Euler      

Derivative as tangential slope Bearbeiten

 
For   the secant slope   converges to the tangential slope. The derivative   is equal to the tangent slope of the tangent touching the graph at the point  .
 
If a differentiable function is used, a tangent can be fitted to it at every m point of the graph. The derivative corresponds to the slope of this tangent.

The derivative   corresponds to the limit value  . The difference quotient   is the slope of the secant between the points   and  . In the case of the boundary value formation  , this secant merges into the tangent that touches the graph of   at the point  :

 
function with a secant and a tangent

Damit ist die derivative   gleich der Steigung der Tangente am Graphen durch den Punkt  . Die derivative kann also genutzt werden, um die Tangente an einem Graphen zu bestimmen. Somit löst sie auch ein geometrisches Problem. Mit   kennen wir die Steigung der Tangente and with   einen Punkt auf der Tangente. Damit können wir die functionsgleichung dieser Tangente bestimmen.

Thus the derivative   is equal to the slope of the tangent to the graph through the point  . we may also use the derivative to compute the tangent to a graph. With   we know the slope of the tangent. The offset can be determined using that   is a point on the tangent. The following question illustrates how this works:

Question: What is the tangent equation if its slope is   and it passes through the point  ?

The general formula of a linear function   is  . Where   is the slope of   and   is the intersection of   with the y-axis (offset). Now let   be the tangent you are looking for. It has slope   and therefore  .

So we only need to find the offset   . Since   passes through the point   , there is

 

So

 

We note: knowing the derivative   at a point (and the point itself) suffices for computing the equation of the tangent.

Derivative as characterization of best approximations Bearbeiten

Approximating a differentiable function Bearbeiten

The derivative can be used to approximate a function. One may even define the derivative as the "best linear approximation" to a function. To find this approximation we start with the definition of the derivative as a limit:

 

The difference quotient   gets arbitrarily close to the derivative  , if   gets sufficiently close to  . For   we can write:

 

In the following we assume, that the expression   for "  is approximately as large as  " is well defined and obeys the common arithmetic laws for equations. So we can change this equation to

 

If   is sufficiently close to  , then   is approximately equal to  . This value can thus be used as an approximation of   near the derivative position. The function with the assignment rule   is a linear function, since   is an arbitrary but fixed point.

The assignment rule   describes the tangent, which touches the graph of the function at the position where the derivative is taken. Thus, the tangent near the point of contact is a good approximation of the graph. This is also shown in the following diagram. If one zooms in close enough at a point in a differential function, the graph looks approximately like a straight line:


 
Differentiable functions locally look like a line

This line is described by the assignment rule   and corresponds to the tangent of the graph at this position.

Example: The sine for small angles Bearbeiten

Let's take a look at the above mentioned example. For this we consider the sine function  . Its graph is

 
The graph of a sine function

As we shall see, the derivative of the sine is the cosine and thus

 

the linear approximation of the sine is hence

 

In the vicinity of zero, there is  . This is the so called small-angle approximation. Thus,   can be approximated by  . With   this approximation is also quite good. The following diagram shows that near zero, the sine function can be described approximately by a line  :

 
Small-angle approximation for the sine function

The diagram also shows that this approximation is only good near the derivative point. For values   far away from zero,   differs greatly from  . The approximation   is therefore only meaningful for small arguments!

Quality of approximations Bearbeiten

How good is the approximation  ? To answer this, let   be the value with

 

The value   is therefore the difference between the difference quotient   and the derivative  . This difference disappears in the limit  , because for this limit the difference quotient turns into a differential quotient, i.e. the derivative  . There is also  . Now we can rearrange the above equation and get

 

The error between   and   is thus equal to the term  . Because of   there is

 

So the error   disappears for  . But we can say even more:   decreases faster than a linear term towards zero. Even if we divide   by   and thus greatly increase this term near  , then   disappears for  . There is

 

The error   in the approximation   thus falls off to zero faster than linear for  . Let us summarize the previous argumentation in one theorem:

Theorem (Approximation of a differentiable function)

Let   and let   be an accumulation point of  . Let also   be differentiable at the point   with the derivative  . Let   and   be defined such that for all   there is


 

Then the error term   for   vanishes, i.e.  . For   there is accordingly  .

Alternative definition of the derivative Bearbeiten

The fact that differentiable functions can be approximated by linear functions characterises the derivative. Every function   is differentiable at the position  , if a real number   (best approximation parameter) as well as a function   exist, such that that   and   apply. Its derivative is then  . There is

 

So we can also define the derivative as follows:

Definition (Alternative definition of the derivative)

Let   be a function and   an accumulation point of  . The function   is differentiable with the derivative   at the point   if a function   exists, such that

 

and   holds.

Describing derivatives using a continuous function Bearbeiten

There is a further characterisation of derivative. We start with the formula

 

Where   is the difference between the difference quotient and the derivative (which disappears for  ). If we rearrange this formula we get:


 

The function   for   has the property

 

Thus   can be extended to a function which is continuous at the position  , whereby the function value is set  . This representation of a differentiable function allows a further characterisation of continuous functions:

Theorem (Equivalent characterisation of the derivative)

A function   is differentiable in   if and only if there is a function   continuous in   with:

 

In that case,  .

Proof (Equivalent characterisation of the derivative)

Proof step:   with       with  

Let  , where   is a function with  . Now, for   there is

 

We now define  . Then we get

 

So   is continuous in   with  .

Proof step:   with       with  

Let now   with a function   continuous in   , where  . For   there is then

 

Now, we define   and get

 

Derivative as generalized slope Bearbeiten

 
The slope of a linear function is given by the quotient  .

The slope is initially only defined for linear functions   with the assignment rule   where  . For such functions the slope is equal to the value   and can be calculated using the difference quotient. For two different arguments   and   from the domain of definition   there is:

 

Now   is also the derivative of   at every accumulation point   of the domain of definition:

 

The derivative of a linear function is therefore always equal to its slope. But the derivative is more general: it is defined for all differentiable functions. (Remember: A term   is a generalisation of another term  , if   is the same as   in all cases where   is defined and   can be applied to other cases.)

So we can consider the derivative as the slope of a function at a point. The transition slope   derivative thus changes from a global property (the slope for linear functions is defined for the whole function), to a local property (the derivative is the instantaneous rate of change of a function).

Examples Bearbeiten

Example of a differentiable function Bearbeiten

Example (The square function is differentiable at  )

 
The graph of the square function
 
The derivative of the square function is given by  

The square function   can be differentiated at the position   with derivative  . We get this result if we evaluate the differential quotient at the position  :

 

The latter expression shows that the difference quotient is equal to   for   (for   the difference quotient is not defined because otherwise we would divide by zero). Now we have to determine the limit value of   as  :

 

Thus the derivative of   at the position   is equal to  , i.e.  . Analogously, we can determine the derivative of   at any position  :

 

Thus the derivative of the square function at the position   is equal to  . The derivative function of   is therefore the function  .

Example of a non-differentiable function Bearbeiten

Example (Absolute value function is not differentiable)

 
The absolute value function is not differentiable at  

We consider the absolute value function   and check whether it can be differentiated at the position  . Here we select the sequences  ,   and   with

 

These all converge to  . What are the differential quotients corresponding to those sequences? For   there is:

 

For   we get:

 

For   there is:

 

This limit for the sequence   does not exist. We therefore see that depending on the sequence   chosen, the limit value   is different or does not exist. Thus, according to definition, the limit value   does not exist either. So the function   cannot be differentiated at the position  . The absolute value function has no derivative at zero.

Left-hand and right-hand derivative Bearbeiten

Definition Bearbeiten

The derivative of a function   is the limit of the difference quotient   for  . The difference quotient can be understood as a function  , which is defined for all   except for  . So   is actually the limit value of a function.

The terms "Left-hand and right-hand derivative" can also be considered for the difference quotient. Thus we obtain the terms "left-hand" and "right-hand" derivative. For the left-hand derivative, only secants to the left of the considered point are evaluated. So only difference quotients   are considered, where  . Then it is checked whether the difference quotient converges to a number in the limit   converge against a number. If the answer is yes, then this number is the left-hand derivative at that point:

 

Here   is the notation for the left-hand derivative of   at the position  . For this limit to make sense, there must be at least one sequence   of arguments that converges from the left towards  . So   has to be an accumulation point of the set  .

Definition (Left-hand derivative)

Let   be a function and   an accumulation point of the set  . The number   is the left-hand derivative of   at the position  , if there is

 

This is equivalent to the statement that for all sequences   from   with   and   and   there is

 

Analogously, the right-hand derivative can be defined as follows:

Definition (Right-hand derivative)

Let   be a function and   an accumulation point of the set  . The number   is the right-hand derivative of   at the position  , if there is

 

This is equivalent to the statement that for all sequences   from   with   and   and   there is

 


functions only have a limit value at one position in their domain of definition if both the left-hand and the right-hand limit value exist at this position and both limit values match. We can apply this theorem directly to derivative functions:

A function is differentiable at a position in its domain of definition if and only if both the left-hand and the right-hand derivative exist there and both derivatives coincide.

Example Bearbeiten

We have already shown that the absolute value function   is not differentiable at  . However, we can still show that the right-hand derivative exists at this position and is equal to  :

 

Analogously, we can show that the left-hand derivative is equal to   at this position:

 

Since the right-hand and left-hand derivatives do not coincide, the absolute value function cannot be differentiated at  . At this point, it has left-hand and right-hand derivatives, but no general derivative.

Weil die rechtsseitige and die linksseitige derivative nicht übereinstimmen, ist die Betragsfunktion an der Stelle   nicht ableitbar. Sie besitzt dort zwar links- and rechtsseitige derivativeen, aber keine derivative.

Differentiable functions do not have kinks Bearbeiten

In the above example we have seen that the absolute value function is not differentiable. This is because the absolute value function "has a kink" at the position  , so that the left-hand and right-hand derivative are different. If we go to   from the left-hand side, the derivative is equal to  , while the derivative from the right-hand side is equal to  . The kink in the absolute value function thus prevents differentiability.

So if a function has a kink, it is not differentiable at this point. In other words: differentiable functions are kink-free. Therefore they are also called smooth functions (actually, smooth means "infinitely many times differentiable"). This does not mean, however, that kink-free functions are automatically differentiable. As an example, let us consider the sign function   with the definition

 

Its graph is

 
Graph of the sign function

This function is not differentiable at the zero point  , because near the the "jump" of the function, the difference quotient converges towards infinity. For the right-hand derivative there is for example:

 

The sign function has no kink at the zero point. Instead, it makes a "jump" there.

At the example of the sign function we see that being "free of kinks" and "differentiable" cannot be the same. However, freedom from kinks is a prerequisite for differentiability. So differentiable functions are free of kinks.

Relations between differentiability, continuity and continuous differentiability Bearbeiten

Continuous differentiability of a function   implies its differentiability, which in turn implies its continuity. The converse statements do not hold, as we will see in the course of this section:

 

The first implication follows directly from the definition: A function   is called continuously differentiable if it is differentiable and the derivative function   is continuous. Thus, continuously differentiable functions are also differentiable. The second implication needs some more work:

Differentiable functions are continuous Bearbeiten

We now show that every at one point differentiable function is also continuous at this point. Thus, differentiability is a stronger condition for a function than continuity:

Theorem

Let   with   be a function, that is differentiable at   . Then,   is also continuous at   . Consequently, every differentiable function   is continuous.

Proof

Let   be any sequence in   converging to  . Since   is differentiable in   , there is a function   ("approximation error") with  , such that for all   in   we have

 

In this case, we will also have   . Since   , we will also have   . So there is:

 

We were allowed to pull the limits apart here because the limits  ,   and   exist. According to the sequence definition for continuity,   implies that   is continuous at  .

Alternative proof

Let   be a sequence in   converging towards   and whose sequence elements are not equal to  . There is also   and   for all  . Since   is differentiable in  , there is  . The derivative of   in the point   is a real number. Then, there is:

 

We were allowed to pull the limits apart here because the limit values   and   exist. Thus   as long as the sequence   attains the value   at most a finite number of times and   holds.

Let now   be any sequence in   which converges towards   and whose sequence elements infinitely often attain the value  . In this case, we take the subsequence of   with sequence elements unequal to   and also obtain the function value limit  . The partial sequence of elements   is constant and its function values trivially converge to  . Thus the sequence   can be split into two subsequences, both of which converge towards  . So we have  .

Hence, for every sequence   in   which converges towards  , there is  . So   is continuous at the position  .

Application: Non-continuous functions are not differentiable Bearbeiten

From the previous section we know that every differentiable function is continuous:

 

Applying the principle of contraposition to this implication, we also get:

 

Example: Non-continuous functions are not differentiable Bearbeiten

Take, as an for example the sign function

 

It is not continuous at  . So it is also not differentiable there. We can prove non-continuity by taking a sequence  . This sequence converges towards zero. If the sign function was differentiable, then the limit value   would have to exist. However

 

The limit value does not exist in  . Therefore the sign function is - as expected - not differentiable at  .

Not every differentiable function is continuously differentiable Bearbeiten

In the following example, we already use some derivatives rules, which will be discussed in more detail in the next chapter. Perhaps you already know them from school. If not, they are a useful insight to what will follow.

Example (Example of a differentiable, but not continuously differentiable function)

We will show that the following function is differentiable everywhere, but its derivative function is not continuous:

 

At   , the product and chain rule (which we will derive later) tells us that the function is infinitely often continuously differentiable. However, at   there is

 

So   is also differentiable at   with derivative  . However, the derivative function   is not continuous at   . To show this, we have to determine the derivative function. For   , the product and chain rule yield

 

Together with the derivative value   we get the derivative function

 

To show the discontinuity of   at   we use the sequence definition of continuity. Let us take the sequence   with  . There is  . If   was continuous, then according to the sequence criterion,   should apply. But now

 

The limit value   does not exist, because the sequence   has the two accumulation points   and  . It follows that   is not continuous at  . Therefore,   is differentiable, but not continuously differentiable.

Exercises Bearbeiten

Hyperbolic function Bearbeiten

Exercise (Hyperbolic function is differentiable at 2)

Show that the hyperbolic function   is differentiable at   and calculate the derivative there. What is the derivative of   at any position  ?

Solution (Hyperbolic function is differentiable at 2)

Here is the differential quotient at the position   is:

 

So   is differentiable at   with the derivative  . For a general   there is

 

Root function Bearbeiten

Exercise (Root function is not differentiable at 0)

Show that the root function

 

is not differentiable at  .

Solution (Root function is not differentiable at 0)

We must show that the differential quotient of   in   does not exist. This quotient is

 

We choose the positive sequence   converging to 0. For this sequence there is

 

Thus there is no limit to the differential quotient  . The function   is therefore not differentiable at  .

Determining limits Bearbeiten

Exercise (Determining limits with differential quotients)

Let   be differentiable in  . Show that the following limits hold:

  1.  
  2.  
  3. Does the reverse statement also hold for the limit value  ? I.e. if the limit value   exists, then   is differentiable at  , and   is equal to this limit?

Solution (Determining limits with differential quotients)

Solution sub-exercise 1:

Since   is differentiable in   , there is

 

If we substitute  , then there is  . Hence

 

Solution sub-exercise 2:

Here, we have

 

Solution sub-exercise 3:

The converse is not true. To show this we consider the function   in  . For this function we have the limit value

 

However, the absolute function is not differentiable at 0.

Criterion for differentiability Bearbeiten

Exercise (Criterion for differentiability of a general function at zero)

Let  . Show: if   for some  , then   is differentiable at 0 with with  .

Solution (Criterion for differentiability of a general function at zero)

There is

 

Since  , there is

 

The squeeze theorem then implies