Understanding Engineering Mathematics

z

0

x

(x 0 ,y 0 ,z 0 )

Plane y = y 0

z = f(x,y)

y

Tangent line with slope∂z

∂x

Figure 16.5Definition of the partial derivative

∂z

∂x

.

The last rule is the generalisation of the function of a function rule (235

➤
), which may
look a little strange in the new notation – again it helps to think of the undifferentiated
variable (yin this case) as a constant.
Similar results apply for the partial derivatives with respect toy.

Problem 16.1

Evaluate the partial derivatives

(i)z=

y

x

Y

x

y

(ii)z=ysin. 2 xY 3 y/

(i) Treatingyas a constant and differentiating with respect toxgives

∂z

∂x

=−

y

x^2

+

1

y

Similarly, keepingxconstant and differentiating with respect toygives

∂z

∂y

=

1

x

−

x

y^2

(ii)

∂z

∂x

= 2 ycos( 2 x+ 3 y)and

∂z

∂y

=sin( 2 x+ 3 y)+ 3 ycos( 2 x+ 3 y)

Exercise on 16.3

Findfx,fyfor the following functions

(i) f(x,y)=

xy

x+y

(ii) f(x,y)=e^3 x+cos(xy)

Answer

(i)fx=

y^2

(x+y)^2

,fy=

x^2

(x+y)^2

(ii)fx=e^3 x+cos(xy)( 3 −ysin(xy)),fy=−xe^3 x+cos(xy)sin(xy)