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Higher Order Partial
Derivatives

Just as we had higher order derivatives with functions of one variable we will also have higher
order derivatives of functions of more than one variable
...

Consider the case of a function of two variables, f ( x, y ) since both of the first order partial
derivatives are also functions of x and y we could in turn differentiate each with respect to x or y
...
Here they are and the notations that we’ll use to denote them
...
Note as well that the order
that we take the derivatives in is given by the notation for each these
...
g
...
In other words, in
this case, we will differentiate first with respect to x and then with respect to y
...
In these cases we differentiate moving along the
fractional notation, e
...

∂y∂x
denominator from right to left
...

Let’s take a quick look at an example
...

Solution
We’ll first need the first order derivatives so here they are
...


+ 6y

f xx = −4 cos ( 2 x ) − 2e5 y
f xy = −10 xe5 y
f yx = −10 xe5 y
f yy = −25 x 2e5 y + 6
Notice that we dropped the ( x, y ) from the derivatives
...
We will also be dropping it for the first order
derivatives in most cases
...
This is not by coincidence
...
So, what’s “nice enough”? The following theorem
tells us
...
If the functions f xy and f yx
are continuous on this disk then,

f xy ( a,b ) = f yx ( a,b )

Now, do not get too excited about the disk business and the fact that we gave the theorem is for a
specific point
...


Example 2 Verify Clairaut’s Theorem for f ( x, y ) = xe

− x2 y 2


...


f x ( x, y ) = e

− x2 y 2

2 − x2 y 2

− 2x y e
2

f y ( x, y ) = −2 yx e

3 − x2 y 2

Now, compute the two fixed second order partial derivatives
...

So far we have only looked at second order derivatives
...
Here are a couple of the third order partial derivatives of function of two
variables
...
There is also another third order partial derivative in which we can do this, f x x y
...
For instance,

f x z ( x, y , z ) = f z x ( x, y , z )

provided both of the derivatives are continuous
...
The
only requirement is that in each derivative we differentiate with respect to each variable the same
number of times
...

Let’s do a couple of examples with higher (well higher order than two anyway) order derivatives
and functions of more than two variables
...

(a) Find f x x y z z for f ( x, y, z ) = z 3 y 2 ln ( x ) [Solution]
∂3 f
xy
for f ( x, y ) = e
[Solution]
(b) Find
2
∂y∂x
Solution
(a) Find f x x y z z for f ( x, y, z ) = z 3 y 2 ln ( x )
In this case remember that we differentiate from left to right
...


z3 y2
fx =
x
z3 y2
f xx = − 2
x
3
2z y
f xxy = − 2
x

6z2 y
f xxyz = − 2
x
12 zy
f xxyzz = − 2
x
[Return to Problems]

∂3 f
xy
for
f
(
x
,
y
)
=
e
(b) Find
∂y∂x 2
Here we differentiate from right to left
...


∂f
= ye xy
∂x
2
∂ f
2 xy
=
y
e
2
∂x

∂3 f
xy
2 xy
=
2
y
e
+
xy
e
2
∂y∂x
[Return to Problems]

The integral is then,

∫∫ F idS = ∫∫∫ div F dV
S

E

2π

⌠
=⎮ ⌠
⎮ ∫
⌡0 ⌡0 0
2π

=⌠
⌡0

1

∫

1
0

4 −3r 2

r dz dr dθ

4r − 3r 3 dr dθ

2π

⌠ ⎛ 2 3 4⎞
= ⎮ ⎜ 2 r − r ⎟ dθ
4 ⎠0
⌡0 ⎝
2π

5
⌠
=⎮
dθ
⌡0 4
5
= π
2

1



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