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
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
is replaced, and as acceleration at the time t is understood.
The letter a comes from acceleratio the Latin word for acceleration (engl. acceleration).
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:
Is f ntimes differentiable on A., so is .
proof by induction over n:
[7.8.2] is mostly used in special cases:
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]:
And trivially, the following applies:
The following remark shows that all cases are real subsets.
Comment: For, applies:
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.
3. ► We prove the claim by induction: For each n there is a function whose n-th derivative is discontinuous.
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