Problem (Chain Rule, 5 Points)
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Let
be continuously differentiable functions. Prove that
and
apply.
We recall (Fischer/Lieb, Page 21 at the bottom): For a differentiable function
, the partial derivatives with respect to
and
are characterized as follows: Let
be continuous functions such that
so is
and
.
We will use this description of the Wirtinger derivatives. Let
. There
in
is differentiable , we have continuous functions
so that
applies.This means
Now set
Da
is
differentiable ,there exist continuous functions
,
so,that
if we insert, this results in
Da
and
of cotinous functions are continuous,is
partially differentiable and
Continuing above,this lead to
and claimed. Analogously follows
Translation and Version Control
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