* *dz z dx z dy

dt x dt y dt

zx

t

y

DIFFERENTIATION OF COMPOSITE FUNCTION

Let z = f ( x, y)Possesses continuous partial derivatives and

let x = g (t) Y = h(t)

Possess continuous derivatives

* *z z

dz dx dyx y

CHANGE OF VARIABLES

Let z=f(x,y)......................(1)

Possess continuous first order partial derivatives w.r.t. x,y.

Let x = (u,v) and y = (u,v)

Possesses continuous first order partial derivatives.

= . . .......(2)z z x z y

u x u y u

Z

X Y

vu = . . ......(3)z z x z y

v x v y v

Differentiation of Implicit Function

Let f(x,y) = 0 or constant number define y as a function

of x implicitly.We shall obtain the value of in terms

of the partial derivatives and .

dy

dxf f

x y

Since f(x,y) is a function of x and y and y is function of x,

therefore we can look upon f(x,y) as a composite function of x.

f f = . .

df dx dy

dx x dx y dx

f f

0 . ...........................( )

-

dyi

x y dx

dy f x

dx f y

1 yExample 1: If Z = tan , prove that

x

Solution: We know that,

dz = . .z zdx dy

x y

2 2

2 2

1 But, .

1

=x + y

z y

y xy x

y

2 2

x dy - y dx dz =

x + y

2 2 2

1and =

x + y1

z x x

y y x

2 2 2 2

2 2

dz = x + y x + y

xdy - ydx =

x + y

y xdx dy

2 2 2

2: Find dz/dt when

z = xy , x=at , y = 2at

Example

x y

2 2

2 2

Solution. We have

z = xy

2 and 2

x y

z zy xy x xy

x y

2 and 2

= . .

dx dyat a

dt dtdz z dx z dy

dt x dt y dt

2 2( 2 ).2 (2 ).2y xy at xy x a

2 2 2 3 2 3 2 4(4 4 ).2 (4 ).2a t a t at a t a t a

3 3 4(16 10 )a t t

2 2

Example 3:

If z= x and y= z , then find the differential co-

efficient of the first order when x is the independent variable.

y x

2

Solution: dz=

Since z = x 2 , 1

z zdx dyx y

z zy x

x y

Thus, dz = 2x dx + dy = 2x dx +dx + 2z dz

dz (1-2z) = dx (2x+1)

dz (2x+1)

dx (1-2z)

, 2 (2 ) (1 4 ) 2

(1 2 ) (1 4 )

(1 4 )

(1 2 )

Also dy dx z xdx dy dx xz zdy

dy z dx xz

dy xz

dx z

Example 4: z is a function of x and y, prove that if x = eu + e-v, y = e-u + e-v then

z z z zx y

u v x y

Solution: z is a change of variable case

. .z z x z y

u x u y u

. .u uz z ze e

u x y

Subtracting, we get

u v u vz z z ze e e e

du dv dx dy

. .z z x z y

v x v y v

= - . .v vz ze e

x y

= xz zy

dx dy

Example 5: If z = ex sin y, where x = In t and y = t2, then find

dz

dt Solution: We know that,

. .dz z dx z dy

dt x dt y dt

sin ,xze y

x

cos ,xze y

y

1

and 2dx dy

tdt t dt

2 = (sin 2 cos )xe

y t yt

1 sin . ( cos )2x xdz

e y e y tdt t

Example 6: If H = f(y-z, z-x, x-y), prove that

0H H H

x y z

Solution: Let, u = y-z, v = z-x, w = x-y → H = f(u,v,w)H is a composite function of x,y,z. We have,

. . .H H u H v H w

x u x v x w x

= .0 .( 1) .1H H H

u v w

= - H H

v w

Similarly

H H H

y w u

H H H

z u v

Adding all the above, we get

0H H H

x y z

Example 7: If x = r cosθ, y = r sinθ and V=f(x,y), then show that

2 2 2 2

2 2 2 2 2

1 1.

V V V V V

x y r r r r

Solution: We have, x = r cosθ, y = r sinθ

2 2 2 1 that and =tany

so r x yx

2 2 cos r r x

r xx x r

sinr y

y r

1cos

x

y r r

2

1sin

y

x r r

therefore V V r V

x r x x

1cos sin

V V

r r

1=cos . sin

V V

r r

1 V= cos . sin Vx r r

1

= cos . sinx r r

. . V V r V

y r y y

1 or = sin . cos

y r r

1=cos . sin

V V

r r

1 = sin . cos V

r r

2

2

V V

x x x

1

cos . sin .r r

1

cos . sin .v v

r r

22

2=cos

V

r

21 sin .cos .

V

r r

21sin .cos .

V

r r

2sin V

r r

22

2 2

1sin

V

r

2

12 sin .cos

V

r

cos . cos .v

r r

1cos . sin .

v

r r

1sin . cos .

v

r r

1 1sin . sin .

v

r r

Adding the result, we get2 2

2 2

V V

x y

2

2

V

y

2

22

=sinV

r

21sin .cos .

V

r r

21sin .cos .

V

r r

2cos V

r r

22

2 2

1cos

V

r

2

12 sin .cos

V

r

V

y y

1sin . cos .

r r

1sin . cos .

v v

r r

2

2 22

cos sinV

r

2

2 22 2

1cos sin

V

r

2 21+ cos sin

V

r r

2 2

2 2 2

1 1 = . .

V V V

r r r r

2 2

2 2 2 22 2

2 2 2 2

8 : If u = x and v = 2xy and f (x,y) = (u,v)

then show that 4

Q y

f fx y

x y u v

Sol: We have

2 and = - 2 y u u

xx y

2 and = 2 x v v

yx y

2

22*2

fx y x y

x u v u v

2 2 2 2

2 22 2 2

4 2f

x xy yx u u v v

2 2 as f (x,y) = (u,v)x yx u v

2 2f

x yx u v

We havef u v

x u x v x

2

22 2

fy x y x

y u v u v

2 2 2 2

2 22 2 2

4 2f

y xy xy u u v v

2 as f (x,y) = (u,v)y xy u v

2 2y xu v

again we have f u v

y u y v y

2 2 2 2

2 22 2 2 2

4f f

x yx y u v

2 2 2 2 22 2

2 2 2 2

2 2 22 2

2 2

Adding the result we get

4 2

4 2

f fx xy y

x y u u v v

y xy xu u v v

1. If z = xm yn, then prove that

dz dx dym n

z x y

2. If u = x2-y2, x=2r-3s+4, y=-r+8s-5, find /u r 3. If x=r cosθ, y=r sinθ, then show that (i) dx = cos θ.dr - r sin θ.dθ (ii) dy = sin θ.dr + r.cos θ.dθ Deduce that (i) dx2 + dy2 = dr2 + r2dθ2

(ii) x dy – y dx = r2.dθ

4. If z = (cosy)/x and x = u2-v, y = eV, find /z v

5. If z=x2+y and y=z2+x, find differential co-efficients of the first order when(i) y is the independent variable.(ii) z is the independent variable.

Exercise

6. If sin cos cos

, , /cos sin sin

u y xz u v find z x

v x y

7. If 1tan log , , .ty dz

z where x t y e findx dt

8. If u = (x+y)/(1-xy), x=tan(2r-s2), y=cot(r2s) then find

9. If z=x2-y2, where x=etcost, y=etsint, find dz/dt.

10. If z=xyf(x,y) and z is constant, show that

'( / ) [ ( / )]

( / ) [ ( / )]

f y x x y x dy dx

f y x y y x dy dx

11. Find and if z = u2+v2+w2, where

u=yex, y=xe-y, w=y/x.

/z x /z y

12. If z=eax+byf(ax-by), prove that 2 .z z

b a abzx y

13. If 2 21 1x y y x a , show that

2

3/22 21

d y a

dx x

14. Find dy/dx if (i) x4+y4=5a2axy. (ii) xy+yx=(x+y)x+y