APPLICATIONS OF

PARTIAL DERIVATIVES

Chapter 3

DIFFERENTIABILITY AND

THE TOTAL DIFFERENTIAL

Chapter 3 Section 1

OUTLINE

Actual change in ,f x y

Vs.

Approximate change in ,f x y

When is differentiable? ,f x y

Relationship between continuity and

differentiability of ,f x y

If f is a function of two variables x and y,

the increment of f at the point (x0, y0), denoted by

f(x0, y0, x, y) is given by

For brevity, we may denote f(x0, y0, x, y)

by f(x0, y0).

000000 ,, ,yxfΔyΔx,yxfyx,yxΔf

Definition.

Example: If f (x, y) = 2x2 + 5xy – 4y2 , find f(x0, y0).

000000 ,,, yxfyyxxfyxf

2

000

2

0

2

000

2

0

452

452

yyxx

yyyyxxxx

2

000

2

0

2

0

2

0

0000

2

0

2

0

452 24

522

yyxxyyyy

yxyxxyyxxxxx

2

000

2

0

2

0

2

0

0000

2

0

2

0

452484

5555242

yyxxyyyy

yxyxxyyxxxxx

.4855524 2

000

2

0 yyyyxyxxyxxx

If f is a function of two variables x and y, and

the increment of f at the point (x0, y0) can be written as

where 1 and 2 are functions of x and y such that

1 0 and 2 0 as (x, y) (0,0), then f is said

to be differentiable at the point (x0, y0).

yxyyxfDxyxfDyxf 2100200100 ,,,

Definition.

Example: If f (x, y) = 2x2 + 5xy – 4y2, show that f is

differentiable in R2.

Solution:

We need to show that

yxyyxfDxyxfDyxf 2100200100 ,,,

where 1 and 2 are functions of x and y such that

1 0 and 2 0 as (x, y) (0,0).

If f (x, y) = 2x2 + 5xy – 4y2 , then

yxyxfD 54,1 00001 54, yxyxfD

yxy,xfD 852 00002 85 yxy,xfD

.4855524 2

000

2

0 yyyyxyxxyxxx

000000 ,,, yxfyyxxfyxf

Theorem.

If a function f of two variables is differentiable at a point, then it is continuous at that point.

(Differentiability implies continuity.)

Remarks:

1) The converse of this theorem is not always true. A

function which is continuous at a point may not be

differentiable at that point.

2) But if a function is not continuous at a point then it is

not differentiable at that point.

10

Theorem.

Let f be a function of x and y such that

D1 f and D2 f exist on an open disk B(P0; r).

If D1 f and D2 f are continuous at P0 , then

f is differentiable at P0.

Example: Show that the indicated function is

differentiable at all points in its domain.

22

3 a.

yx

xy,xf

11

The domain of f is the set of

points in R2 except (0,0).

222

22

1

233

yx

xxyxy,xfD

222

22 33

yx

xy

222

222 633

yx

xyx

222

22

2

230

yx

yxyxy,xfD

222

6

yx

xy

Since D1f and D2f are rational functions, each is

continuous at every point in R2 except at (0,0).

Hence f is differentiable at all points in its domain.

Solution:

The domain of g is . 02 :xRx,y

x

yyxg x , xyxg y ln, and

Since gx is continuous at the point (x, y) whenever

x 0, and gy is continuous at the point (x, y)

whenever x > 0, then g is differentiable at all

points in its domain. 12

xyyxg ln, b.

00 if0

00 if3

44

22

.

,y,x

,y,xyx

yx

y,xfExample: Let

Show that and exist but f is

not differentiable at (0, 0).

0,01 fD 0,02 fD

solution:

0

0,00,lim0,0

01

x

fxffD

x

xx

00lim

0

xx

0lim

0

0

0

0,0,0lim0,0

02

y

fyffD

y

yy

00lim

0

yy

0lim

0

013

Take S1 as the set of points on the y- axis and S2

as the set of all points on the line given by y = x.

Therefore f is discontinuous at (0,0) and hence

f is not differentiable at (0,0).

does not exist.

14

44

22

0000

3lim,lim

11

yx

yxyxf

S x,y,x,y

S x,y,x,y

00lim0

y

40

0lim

yy

44

22

0000

3lim,lim

22

yx

yxyxf

S x,y,x,y

S x,y,x,y

4

4

0 2

3lim

x

x

x

2

3

2

3lim

0

x

44

22

0,0,

3lim

yx

yx

yx

If f is a function of two variables x and y,

and f is differentiable at (x, y), the total

differential of f is the function df such that

15

yyxfDxyxfDyxyxdf ,,,,, 21

Definition.

y,x,,df 12 a. 02001012 b. .,.,,df

Example: If f (x, y) = 2x2 + 5xy – 4y2,

Find:

solution:

yy,xfDxy,xfDy,x,y,xdf 21

yyxxyxy,x,y,xdf 8554

yxy,x,,df 1825152412

yx 81058

yx 183

02018010302001012 ...,.,,df

330360030 ... 16

y)y,x(fDx)y,x(fDdz 21 1.

dy)y,x(fDdx)y,x(fDdz 21 2.

dyy

zdx

x

zdz

3.

Remarks: If z = f (x, y), then

17

AfxaxaxafAf nn ,....,, 2211

If f is a function of n variables ,

and A is the point , the increment of f

at A is given by

nxxx ,..,2,1

naaa ,..,2,1

Definition.

where

then f is said to be differentiable at A.

If f is a function of n variables ,

and the increment of f at the point A can be

written as

nxxx ,..,2,1

nn

nn

xxx

xAfDxAfDxAfDAf

...

...

2211

2211

,0,...,0,0,...,, as ,0,...,0,0 2121 nn xxx

Definition.

If f is a function of n variables ,

and f is differentiable at a point A, the total

differential of f is given by

nxxx ,..,2,1

....,....,,, 221121 nnn xAfDxAfDxAfDxxxAdf

Remark:

If w is a function of n variables then

nxxx ,..,2,1

n

n

dxx

wdx

x

wdx

x

wdw

...2

2

1

1

Definition.

AfxaxaxafAf nn ,....,, 2211

nn

nn

xxx

xAfDxAfDxAfDAf

...

...

2211

2211

....,....,,, 221121 nnn xAfDxAfDxAfDxxxAdf

When f is differentiable and ∆xi is “small” for i = 1 to n,

then dff

22

23