The Chemistry Maths Book, Second Edition

9.3 Partial differentiation 251

EXAMPLES 9.2Partial differentiation

(i) f(x,y,z) 1 = 1 x

2

1 + 12 y

2

1 + 13 z

2

1 + 14 xy 1 + 15 xz 1 + 16 yz

(ii) f(x,y) 1 = 1 (x

2

1 + 12 y

2

)

122

Letf 1 = 1 u

122

whereu 1 = 1 x

2

1 + 12 y

2

. Then, by the chain rule,

(iii) f(x,y) 1 = 1 y 1 sin(x

2

1 + 1 y

2

)

By the chain rule,

To find , letf 1 = 1 u 1 × 1 vwhereu 1 = 1 yandv 1 = 1 sin(x

2

1 + 1 y

2

). Then, by the product

rule,

0 Exercises 3–7

Higher derivatives

Like the derivative of a function of one variable (Section 4.9), the partial derivative

of a function of more than one variable can itself be differentiated if it satisfies the

necessary conditions of continuity and smoothness. For example, the cubic function

in two variables

z 1 = 1 x

3

1 + 12 x

2

y 1 + 13 xy

2

1 + 14 y

3

has partial first derivatives

∂

∂

=++,

∂

∂

=++

z

x

xxyy

z

y

343 2612 xxyy

22 2 2

=+++ 2

222 22

yxy xycos( ) sin( )

∂

∂

=×

∂

∂

+×

∂

∂

=× + + +

f

y

u

y

u

y

yy xy xy

v

v 2

22 22

cos( ) sin( ))× 1

∂

∂

f

y

∂

∂

=+

f

x

2 xy x y

22

cos( )

∂

∂

=×

∂

∂

=×=+

−−

f

y

df

du

u

y

uyyxy

1

2

42 2

12 2 2 12

()

∂

∂

=×

∂

∂

=×=+

−−

f

x

df

du

u

x

uxxxy

1

2

22

12 2 2 12

()

∂

∂

=++,

∂

∂

=++,

∂

∂

=++

f

x

xyz

f

y

yxz

f

z

245 446 656 zxy