218 Chapter 6 Let’s Learn About Partial Dierentiation!
Derivatives of Implicit Functions
A point (x, y) for which a two-variable function f(x, y) is equal to constant c
describes a graph given by f(x, y) = c. When a part of the graph is viewed as a
single-variable function y = h(x), it is called an implicit function. An implicit
function h(x) satisfies f(x, h(x)) = c for all x defined. We are going to obtain
h(x) here.
When z = f(x, y), the formula of total differentials is written as dz = f
x
dx +
f
y
dy. If (x, y) moves on the graph of f(x, y) = c, the value of the function f(x, y)
does not change, and the increment of z is 0, that is, dz = 0. Then, we get
0 = f
x
dx + f
y
dy. Assuming f
y
≠ 0 and modifying this, we get
dy
dx
f
f
x
y
=
The left side of this equation is the ideal expression of the increment
of y divided by the increment of x at a point on the graph. It is exactly the
derivative of h(x). Thus,
( )
= h x
f
f
x
y
Example
f(x, y) = r
2
, where f(x, y) = x
2
+ y
2
, describes a circle of radius r centered
at the origin. Near a point that satisfies x
2
r
2
, we can solve f(x, y) = x
2
+
y
2
= r
2
to find the implicit function y = h(x) = r
2
x
2
or
y h x r x=
( )
=
2 2
.
Then, from the formula, the derivative of these functions is given by
( )
= = h x
f
f
x
y
x
y
Exercises
1. Obtain f
x
and f
y
for f(x, y) = x
2
+ 2xy + 3y
2
.
2. Under the gravitational acceleration g, the period T of a pendulum hav-
ing length L is given by
T
L
g
= 2
π
(the gravitational acceleration g is known to vary depending on the
height from the ground).
Obtain the expression for total differential of T.
If L is elongated by 1 percent and g decreases by 2 percent, about
what percentage does T increase?
3. Using the chain rule, calculate the differential formula of the implicit
function h(x) of f(x, y) = c in a different way than above.
Epilogue:
What Is Mathematics for?
220 Epilogue
Phew, it’s hot!
No maer where
they put me, I do
my best.
We, where is
the Asagake
Times Okinawa
Oice?
Naha Airport
What Is Mathematics For? 221
This situation
lks a t
familiar to me!!
You?!?
You aren’t
the head of
this oice,
are you?!?
No way!
I just got
here from
the airport,
t.
Oh, that’s
gd!
Who is in charge
of this oice?
G o n n n g !
Whsh
But you haven’t
bn here long
enough to be
slping already!!
You lazy bum!
Eek!
Trot
Trot
The Asagake Times
Okinawa Office
222 Epilogue
Excuse me, do
you know where
the person in
chargeis?
Oh, he is always
swiing.
There you are!
Pat
Pat
Pat

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