What is a doubly differentiable function

7.8. Multiple differentiable functions


At first glance, the transition from the local to the global aspect of differentiability seems to be reflected only in a more compact notation. But if you consider that you are a derivativefunction - as opposed to a derivativenumber - check again for differentiability, i.e. can possibly generate the functions one after the other, it becomes clear that a new platform has also been gained with the new perspective.

To conceptually grasp the multiple differentiability precisely is technically somewhat complex and only recursive possible.

Definition: It be and. A function is called

  1. Can be differentiated once on A. if f on A. is differentiable. We call the function the 1st derivative of f.

  2. (n +1) times differentiable on A. if f  n-mal and the n-th derivative can be differentiated once A. is. We call the function the (n + 1) -th derivative of f.

[7.8.1]

we read as "f n" or as "f above n"and occasionally speak of it as the 0th derivative. We mostly use it for small ones n the spelling, etc.

denote the set of all n-time differentiable functions, in short: -functions, on A..

A function is called n-time continuously differentiable on A., or a function, if the n-th derivative is continuous. We denote the set of functions with the symbol.

The functions, i.e. the functions

,

lie in everyone. They can therefore be differentiated as often as desired.

:

  • Again to physics and its special notation (cf. [7.3]): In the case of multiply differentiable functions of the form, the point notation is of course also used for the derivatives, with the second derivative usually using the symbol a

    i

    The letter a comes from acceleratio the Latin word for acceleration (engl. acceleration).

    is replaced, and as acceleration at the time t is understood.


     

The recursive character often makes it difficult to prove the higher differentiability of a function. The 4-fold differentiability, for example, only follows from the 3-fold, which in turn presupposes the 2-fold, etc. We show this using the example of the cube function.

Since it is differentiable ([7.3.3]), we get the following results with the factor rule [7.7.10]:

  is 1-times differentiable and
  is 2 times differentiable and
  is 3 times differentiable and
  is 4 times differentiable and

The last piece of information makes it clear that there is even a function whose derivatives are constant 0 from the 4th order.

To a present function f as e.g. to be recognized 10 times differentiable, one has to show according to [7.8.1] that it is differentiable again. But perhaps it is easier with this function to recalculate the nine-fold differentiability of, or the three-fold of. Interestingly, all of these variants lead to the same result.

Comment: Be . The following applies to:

[7.8.2]

Is f  ntimes differentiable on A., so is .

proof by induction over n:

  1. There is nothing to show for because of the condition.

  2. Let the equivalence [7.8.2] and the derivation formula noted there already be valid. We now have to show for:

    as . For this is given directly by the definition [7.8.1], so that we may assume in the following.

    "" Be, i.e., with.
    According to the induction hypothesis, and with

    .

    So you know: and.

    "" Be now, so that.
    According to definition [7.8.1] this means: and. According to the induction hypothesis, and, thus:. This secures.

[7.8.2] is mostly used in special cases:

[7.8.3]

where the calculation is permissible in the case of derivation.
 

There are obvious subsets of relationships between the individual differentiability classes. For example, it follows from [7.8.2]:

[7.8.4]
[7.8.5]

And trivially, the following applies:

[7.8.6]

The following remark shows that all cases are real subsets.

Comment: For, applies:

[7.8.7]
[7.8.8]
[7.8.9]

proof: We consider o.E. just the case. All functions constructed in the following work with 0 as a critical point. By means of a suitable shift, such a critical point can be created in any amount A. establish so that the general situation can also be grasped.

Also note that the function constructed in 2. is also a counterexample to 1..

2. ► First we consider for the function

According to the product rule ([7.6.3], see also [7.4.3] for the derivation of the absolute value function) is in each differentiable with

The differentiability in 0 follows from

So you have: and.

We now come to the actual proof. In doing so, we may limit ourselves to the case and show for by induction: belongs to, but not to, therefore not to either.

  • Be . According to our preliminary considerations, there is a function that has no function.

  • According to the induction hypothesis, there is a - but not a function. According to [7.8.3] this means: is a function that is missing in.

3. ► We prove the claim by induction: For each n there is a function whose n-th derivative is discontinuous.

  • By