218 Chapter 6 Let’s Learn About Partial Dierentiation!

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 maer where

they put me, I’ do

my best.

We, where is

the Asagake

Times Okinawa

Oice?

Naha Airport

What Is Mathematics For? 221

This situation

lks a t

familiar to me!!

You?!?

You aren’t

the head of

this oice,

are you?!?

No way!

I just got

here from

the airport,

t.

Oh, that’s

gd!

Who is in charge

of this oice?

G o n n n g !

Whsh

But you haven’t

bn here long

enough to be

slping already!!

You lazy bum!

Eek!

Trot

Trot

The Asagake Times

Okinawa Office

222 Epilogue

Excuse me, do

you know where

the person in

chargeis?

Oh, he is always

swiing.

There you are!

Pat

Pat

Pat

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