Derivatives of higher order – Serlo

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Motivation[Bearbeiten]

Diagram for location, speed, acceleration and jerk of an object. The location is the turquoise line. The velocity (violet) increases, is then constant between and and then drops back to zero. As soon as the speed drops, the acceleration (green) becomes negative. A jerk only occurs in areas that are not constantly accelerated and is a step function. Therefore, the derivative of the jerk is also zero within the flat pieces (at the jumps, the derivative is not defined).

The derivative describes the current rate of change of the function . Now the derivative function can be differentiated again, provided that it is again differentiable. The obtained derivative of the derivative is called second derivative or derivative of second order and is called or . This can be done arbitrarily often. If the second derivative is again differentiable, a third derivative can be constructed, then a fourth derivative and so on.

These higher derivatives allow statements about the course of a function graph. The second derivative tells us whether a graph is curved upwards ("convex") or curved downwards ("concave"). If a function has a convex graph, its gradient increases continuously. For this convexity, is a sufficient condition. If the second derivative is always positive, then the first derivative must grow continuously. Analogously, it follows from that the graph is concave and the derivative falls monotonously.

Higher-order derivatives do not only tell us more about abstract functions, they can also have a physical meaning. Consider the function with , which shall describe the location of a car at the time . We already know that we can calculate the speed of the car at the time with the first derivative: . What does the derivative of say? This is the instantaneous rate of change of speed and thus the acceleration of the car. It accelerates with . So second derivatives describe accelerations.

Now we can derive this second derivative again, whereby we get the rate of change of acceleration . This is called jerk in vehicle dynamics and indicates how fast a car increases acceleration or how fast it initiates braking. For example, a big jerk occurs during emergency braking. Since is in an emergency stop, the graph of the speed is convex - the speed decreases more and more. The fourth derivative again tells us that the jerk has no instantaneous rate of change.

Definition[Bearbeiten]

Definition (Derivatives of higher order)

Let with be a real function. We set and in the case of differentiability . We define the second derivative via , the third derivative via etc., if these higher derivatives exist. Overall, we define recursively for :

We say that is times differentiable, if the -th derivative of exists. is called times continuously differentiable, if is continuous (which is a stronger statement).


The set of all times continuously differential functions with domain of definition and range is denoted . In particular consists of the continuous functions. If we can derive the function arbitrarily often, we write . If , then we can write or in short. Those sets of functions satisfy the inclusion chain:

Question: Are the following statements true of false?

Solutions:

  1. true
  2. false
  3. false
  4. false
  5. true
  6. true

Examples for higher derivatives[Bearbeiten]

Derivatives of the power function[Bearbeiten]

Example (Derivatives of the power function)

We consider the function . This function is infinitely often differentiable, since there is for all and all :

In general, for with there is:

Derivatives of the exponential function[Bearbeiten]

Example (Derivatives of the exponential function)

For the exponential function since for all we have infinite differentiability . In addition there is for all :

Derivatives of the sine function[Bearbeiten]

Example (Derivatives of the sine function)

The function is infinitely often continuously differentiable. For all there is:

In general, for all there is:

Question: What are the derivatives von ?

We use that . For there is

In general, for all there is:

Exercises: higher derivatives[Bearbeiten]

Derivatives of the logarithm function[Bearbeiten]

Exercise (Derivatives of the logarithm function)

Show that the logarithm function is arbitrarily often differentiable and that for all there is:

Proof (Derivatives of the logarithm function)

Theorem whose validity shall be proven for the :

1. Base case:

1. inductive step:

2a. inductive hypothesis:

2b. induction theorem:

2b. proof of induction step:

Exactly once differentiable function[Bearbeiten]

Exercise (Exactly once differentiable function)

Prove that the following function is differentiable once, but not twice:

Solution (Exactly once differentiable function)

The function with for and .

This function is differentiable in all points , since for all in the open neighbourhood for or for there is . Consequently, by the product and the chain rule

For we obtain

Since for all there is , so the term is bounded. Hence, for the derivative function

However, this function is not differentiable at . We approach 0 by taking two sequences and , where we define for all

Then, there is and . Further there is for all

So there is

But

Consequently the limit value does not exist and therefore is not differentiable at .

Additional question: Is continuous at ?

Nope. Take the two sequences:

For these sequences, there is: . However

So doesn't exist. By means of the sequence criterion, is hence not continuous at .

Remark: Therefore, is also not differentiable at .

Computation rules for higher derivatives[Bearbeiten]

Linearity[Bearbeiten]

The linearity of derivatives is also "inherited" to higher derivatives: If and are differentiable, for the function is also differentiable with

If and are now even twice differentiable, then there is

If we continue to do so, we will get

Theorem (Linearity of higher derivatives)

Let and be times differentiable. Then also is times differentiable, and for all there is:

Example (Linearity of higher derivatives)

Since and for there is

Proof (Linearity of higher derivatives)

Theorem whose validity shall be proven for the :

1. Base case:

1. inductive step:

2a. inductive hypothesis:

2b. induction theorem:

2b. proof of induction step:

Leibniz rule for product functions [Bearbeiten]

We now try to determine a general formula for the -th derivative of the product function of two arbitrarily often differentiable functions and . By applying the factor-, sum- and product rule several times we obtain for

If we plug in and , and instead of the derivatives of and the corresponding powers of and , we see a clear analogy to the binomial theorem:

This analogy can be made clear as follows:

We assign for every the derivative to the power , and the derivative to the power . The -th derivative corresponds to the -th power . The derivative of the term is by means of the product rule

The expression now corresponds in our analogy to the sum . We get this term from by multiplication with . For our polynomials, the distributive law yields

Therefore, the application of the product rule corresponds to the multiplication with the sum . Thus the -th derivative corresponds to the power . From the binomial theorem

we hence get the

Theorem (Leibniz rule for derivatives)

Let be times differentiable functions. Then, is times differentiable, and for all , there is:

Example (Leibniz rule for derivatives)

Using the Leibniz rule we calculate . The rule is applicable because and are arbitrarily often differentiable on . There is

Proof (Leibniz rule for derivatives)

Theorem whose validity shall be proven for the :

1. Base case:

1. inductive step:

2a. inductive hypothesis:

2b. induction theorem:

2b. proof of induction step: