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Unformatted text preview: 344 CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
and, if they are also diﬀerentiable, each has two partial derivatives, which
are the secondorder partials of f :
∂2f
∂2x
∂2f
∂y∂x
∂2f
∂x∂y
∂2f
∂2y ∂
∂x
∂
=
∂y
∂
=
∂x
∂
=
∂y
= ∂f
∂x
∂f
∂x
∂f
∂y
∂f
.
∂y In subscript notation, the above would be written fxx = (fx )x
fxy = (fx )y
fyx = (fy )x
fyy = (fy )y .
Notice that in the “partial” notation, the order of diﬀerentiation is
righttoleft, while in the subscript version it is lefttoright. (We shall see
shortly that for C 2 functions, this is not an issue.)
For example, the function f (x, y ) = x2 + 2xy + y − 1 + xy 3
has ﬁrstorder partials
∂f
= 2x + 2y + y 3
∂x
∂f
= 2x + 1 + 3xy 2
fy =
∂y fx = ...
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 Fall '08
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 Calculus, Derivative

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