== Exercise Sheet 5 - Partial Derivatives

=== Exercise 1 @Exercise[5, 1]
#block(
  fill: luma(230),
  inset: 8pt,
  radius: 4pt,
[
	Where is the function
$
	f: RR^2 -> RR

	(x,y) |-> x dot sqrt(y^2 + x^2)
$
	partially differentiable? Calculate the first-order partial derivatives.
])

$
g(x,y) &= x ; (delta g)/(delta x) = 1 ; (delta g)/(delta y) = 0 \
h(x,y) &= sqrt(y^2+x^2) ; (delta h)/(delta x) = sqrt(y^2+x^2) dot 2x ; (delta h)/(delta y) = sqrt(y^2+x^2) dot 2y \
\
(delta f)/(delta x) &= g dot h' + g' dot h = x dot sqrt(y^2+x^2) dot 2x + 1 dot sqrt(x^2+y^2) \
&= sqrt(x^2+y^2) (2x^2 + 1) \

(delta f)/(delta y) &= g dot h' + g' dot h = x dot sqrt(x^2+y^2) dot 2y + 0 dot dots \
&= x dot sqrt(x^2+y^2) dot 2y

\ "dödö falsch"
$
