Derivative of the inverse function[Bearbeiten]
Solution (Derivative of the inverse function)
Part 1:
Proof step: Existence and computation of 
Proof step: Existence and computation of 
Part 2:
Proof step: Computation of 
There is
So
Proof step: Computation of 
Derivative of a general logarithm function [Bearbeiten]
Exercise (Derivative of a general logarithm function)
Show that the function
with base
is bijective and differentiable on
. Show further that the inverse function
is differentiable on all of
with derivative
.
Solution (Derivative of a general logarithm function)
Proof step:
is bijective and differentiable
Fall 1: 
Fall 2: 
Proof step:
exists and is differentiable
Proof (derivatives on
and area-functions)
Differentiability of
:
For the cotangent function
there is:
. Thus the function is differentiable and strictly monotonically decreasing, and thus injective. Further,
. Thus
is bijective. The inverse function
is differentiable according to the theorem on the derivative of the inverse function, and for
there is:
Differentiability of
:
The hyperbolic sine function
is differentiable with
. Thus it is strictly monotonously increasing, and hence injective. Further,
. So it is also surjective. The inverse function
is differentiable by the theorem on the derivative of the inverse function , and for
there is:
Differentiability of
:
The hyperbolic cosine function
is differentiable with
on
. Thus it is strictly monotonically increasing, and hence injective. Further,
. So it is also surjective. The inverse function
is differentiable by to the theorem on the derivative of the inverse function, and for
there is:
Differentiability of
:
For the cotangent function
there is:
. Thus the function is strictly monotonically increasing, and thus injective. Further,
. Thus
is bijective. The inverse function
is differentiable by the theorem on the derivative of the inverse function, and for
there is:
Exercise (Non-differentiable functions at zero)
Let
. Show that:
- Let
with
and
for all
. Then
and
are not simultaneously differentiable at zero.
- Let
, and let
be differentiable at zero. Further let
and
for all
. Then
is not differentiable at zero.
Derivatives of higher order[Bearbeiten]
Exercise (Arbitrarily often differentiable function)
Show that the function
is arbitrarily often differentiable and for all
there is:
Proof (Arbitrarily often differentiable function)
The proof goes by induction over
:
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:
Exercise (Exactly one/two/three times differentiable functions)
Provide an example of a
- function

- function that is differentiable, but not continuously differentiable on

- function

Solution (Exactly one/two/three times differentiable functions)
Solution sub-exercise 1:

or

or
Solution sub-exercise 2:

or
Solution sub-exercise 3:

or

or
Solution sub-exercise 4:

or

or
Hint
We can successively proceed with the construction of functions and obtain some
for all
, with

or

or
Exercise (Application of the Leibniz rule)
Determine the following derivatives using the Leibniz rule

for 
Solution (Application of the Leibniz rule)