1.1. PARTIAL DERIVATIVES 5
As long as we remember that y is the active variable and x is treated as a constant, this is not
difficult.
Example 1.4 Find the partial derivatives with respect to x and y of
f .x; y/ D .3x C 4y/
5
Solution:
f
x
.x; y/ D 5.3x C4y/
4
3 f
y
.x; y/ D 5.3x C4y/
4
4
e only point where things are different is the way the chain rule acts depending on if x or y
is the active variable.
˙
Example 1.5 Find the partial derivatives with respect to x and y for
f .x; y/ D
x
y
Solution:
f
x
.x; y/ D
1
y
f
y
.x; y/ D
x
y
2
For f
x
,
1
y
is effectively a constant, while, for f
y
, we employ the reciprocal rule while x plays the
part of a constant.
˙
1.1.1 IMPLICIT PARTIAL DERIVATIVES
Since natural laws are often stated in the form of equations that are not in functional form,
implicit derivatives are very useful in physics. It turns out that implicit partial derivatives are a
lot like standard partial derivatives – as long as you remember which variable is active.
Example 1.6 Find z
x
and z
y
if x
2
C y
2
C z
2
D 16.
Solution:
z is the dependent variable and so gets a z
x
or a z
y
each time we take a derivative of it,
while x and y take turns being the active variable and a constant, respectively. So,
2x C 2z z
x
D 0 and 2y C2z z
y
D 0
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