The Manga Guide to Calculus

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