3.1 Iterated Partial Derivatives
3.1 Iterated Partial Derivatives
Prof. Tesler
Math 20C Fall 2018
Prof. Tesler
3.1 Iterated Partial Derivatives
Math 20C / Fall 2018 1 / 19
Higher Derivatives
Take the partial derivative of f (x, y) = x2y3 with respect to x:
fx(x, y) = 2xy3
This is also a function of x and y, and we can take another derivative with respect to either variable:
The x derivative of fx(x, y) is ( fx)x = fxx = 2y3. The y derivative of fx(x, y) is ( fx)y = fxy = 6xy2. fxx and fxy are each an iterated partial derivative of second order .
The y derivative of the x derivative can also be written:
(x2y3) = (2xy3) = 6xy2
y x
y
or 2 (x2y3) = 6xy2 y x
Prof. Tesler
3.1 Iterated Partial Derivatives
Math 20C / Fall 2018 2 / 19
Iterated Derivative Notations
Let f (x, y) = x2y3.
There
are
two
notations
for
partial
derivatives,
fx
and
f x
.
Partial derivative of f with respect to x in each notation:
fx = 2xy3
f (x, y) = f = 2xy3
x
x
Partial derivative of that with respect to y:
( fx)y = fxy, so fxy(x, y) = 6xy2
2
f=
f
y x
y x
= 2f = 6xy2 y x
Notice derivatives are listed in the opposite order in each notation.
Prof. Tesler
3.1 Iterated Partial Derivatives
Math 20C / Fall 2018 3 / 19
Iterated Derivative Notations
Notice derivatives are listed in the opposite order in each notation.
In each notation, compute the derivatives in order from the one listed closest to f , to the one farthest from f :
5
5f
fxyzzy = y z z y x f = y z2 y x f = y z2 y x
Both notations say to take derivatives in the order x, y, z, z, y.
Prof. Tesler
3.1 Iterated Partial Derivatives
Math 20C / Fall 2018 4 / 19
Mixed Partial Derivatives
f (x, y) = x2y3
fx = 2xy3 fxx = 2y3 fxy = 6xy2
fy = 3x2y2 fyx = 6xy2 fyy = 6x2y
A mixed partial derivative has derivatives with respect to two or more variables.
fxy and fyx are mixed. fxx and fyy are not mixed.
In this example, notice that fxy = fyx = 6xy2. The order of the derivatives did not affect the result.
Prof. Tesler
3.1 Iterated Partial Derivatives
Math 20C / Fall 2018 5 / 19
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