Understanding Engineering Mathematics

Note also that these error problems can be taken over directly to problems of small
increases – e.g. the expansion of a rectangular box due to heating, etc. (using coefficients
oflinearexpansions).
In fact, we have already met the total differential disguised asimplicit differentiation
(238

➤

).

Problem 16.6

Ifz=x^2 Y 5 x^2 yY 2 xy^2 −y^3 =4 find

dy

dx

.

We fi n d

dy

dx

by noting thatdz=0, using the total differential, and solving the resulting

equation to obtain a relation betweendxanddywhich we then solve for

dy

dx

. Thus

dz= 2 xdx+ 10 xydx+ 5 x^2 dy+ 2 y^2 dx+ 4 xydy− 3 y^2 dy= 0

Dividing bydxgives

2 x+ 10 xy+ 5 x^2

dy

dx

+ 2 y^2 + 4 xy

dy

dx

− 3 y^2

dy

dx

= 0

and solving for

dy

dx

gives

dy

dx

=

2 x+ 10 xy+ 2 y^2

3 y^2 − 5 x^2 − 4 xy

Exercises on 16.5

1. Find the total differentialdzwhen

(i) z=ln(cos(xy)) (ii) z=exp

(

x

y

)

2. Ifz=e^2 x+^3 yandx=lnt,y=t^2 , calculate

dz

dt

from the total derivative formula and

show that it agrees with the result obtained by substitution forxandybefore differ-

entiating.

Answers

1. (i)−tan(xy)(ydx+xdy) (ii)

1

y^2

exp

(

x

y

)

(ydx−xdy)

2. 2t( 3 t^2 + 1 )exp( 3 t^2 )

16.6 Reinforcement

1.Find the values of the following functions at the points given:

(i)f(x,y)= 2 xy^3 + 3 x^2 yat the point (2, 1)

(ii) g(x,y)=(x+y)exsinyat the point (0,π/2)