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
x
dz z dx z dy
 *  *
dt x dt y dt
z
z
dz  * dx  * dy
x
y
y
t
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.
Z
z
z x z y
=
.  . .......(2)
u
x u y u
z
z x z y
=
.  . ......(3)
v
x v y v
X
Y
u
v
Differentiation of Implicit Function
Let f(x,y) = 0 or constant number define y as a function
dy
of x implicitly.We shall obtain the value of
in terms
dx
f
f
of the partial derivatives
and
.
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.
df
f dx f dy

=
. 
.
dx
x dx y dx

f
f dy
0

.
...........................(i )
x
y dx
dy
f x

 dx
f y
 y
Example 1: If Z = tan   , prove that
x
1
x dy - y dx
dz =
2
2
x +y
Solution: We know that,
z
z
dz =
.dx  .dy
x
y
z
1
 y 
But,

.
2  2 
y
1  y x  x 
y
= 2
2
x +y
z
1x
x
and
=
 2
2
2
y 1   y x 
x +y
y
x
 dz = 2
dx

dy
2
2
2
x +y
x +y
xdy - ydx
= 2
2
x +y
Example 2:
Find dz/dt when
z = xy  x y,
2
Solution.
2
2
x=at ,
y = 2at
We have
z = xy  x y
2
z
2
 y  2 xy
x
and
2
z
2
 x  2 xy
y
dx
dy
 2at
and
 2a
dt
dt
dz
z dx z dy

=
.  .
dt
x dt y dt
 ( y  2 xy).2at  (2 xy  x ).2a
2
2
 (4a2t 2  4a2t 3 ).2at  (4a2t 3  a2t 4 ).2a
 a3 (16t 3  10t 4 )
Example 3:
If z= x 2  y and y= z 2  x, then find the differential coefficient of the first order when x is the independent variable.
z
z
Solution:
dz= dx  dy
x
y
z
z
2
Since
z=x y

 2 x,
1
x
y
Thus,
dz = 2x dx + dy = 2x dx +dx + 2z dz
 dz (1-2z) = dx (2x+1)
dz (2x+1)


dx
(1-2z)
Also,
dy  dx  2 z (2 xdx  dy)  dx (1  4 xz)  2 zdy
 dy (1  2 z )  dx (1  4 xz)
dy (1  4 xz )


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
z

x y
u v
x
y
Solution: z is a change of variable case
z
z x

.

u
x u
z
z u

.e 
u
x
z y
.
y u
z  u
.e
y
z
z x
z y

.

.
v
x v
y v
z
z
v
= .e

.e v
x
y
Subtracting, we get
z z z u
z u
v
v

 e  e   e  e 
du dv dx
dy
z
z
=x
y
dx
dy
Example 5: If z =
ex
sin y, where x = In t and y =
t2,
then find
Solution: We know that,
dz
z dx
z dy

.

.
dt
x dt
y dt
z
x
 e sin y,
x
dx 1

dt t
z
x
 e cos y,
y
dy
and
 2t
dt
dz
1
x
x

 e sin y.  (e cos y )2t
dt
t
x
e
2
=
(sin y  2t cos y )
t
dz
dt
Example 6: If H = f(y-z, z-x, x-y), prove that
H H H


0
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
H
H
H
=
.0 
.(1) 
.1
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
H H H


0
x
y
z
Example 7: If x = r cosθ, y = r sinθ and V=f(x,y),
then show that
 V  V  V 1 V 1  V
 2  2  .
 2
2
2
x
y
r
r r r 
2
2
2
2
Solution: We have, x = r cosθ, y = r sinθ
so that r  x  y
2
r
2r
 2x
x
2

r y
  sin 
y r
2
and
 =tan
1
r x
  cos 
x r
y
x
 x 1
  cos 
y r r
V V r V 


x
r x  x
  y 1
 2  sin 
x r
r
therefore
V
V

cos  
r

V 1
V
 1

 sin   =cos . r  r sin  
 r


 1
 

V=  cos .  sin 
V
x
r r
 

 
 1
 
=  cos .  sin 

x 
r r
 
V V r V 

. 
.
y
r y  y
V 1
V
=cos .
 sin 
r r

 1
 

=  sin  .  cos  V
r r
 


or
=
y
 1
 

 sin  .  cos  
r r
 

 1
 
 2V  V 

  cos  .  sin  . 
2
x
x x 
r r
 
v 1
v 

 cos  .  sin  . 
r r
 


v 

1

v


 cos .  cos .   cos  .

sin

.


r 
r 
r  r
 
1

v  1
 1
v 
 sin  .  cos  .   sin  .   sin  . 
r
 
r  r
  r
 
2
2

V
1
V
1
V
2
=cos  2  sin  .cos  .
2 2 sin  .cos 
r
r
r
r

2
2
2
1
V
sin   V
1
V
2
 2 sin 

 sin  .cos .
2
r

r r
r
r
 1
 
v 1
v 
 V
 2V


  sin  .  cos .   sin  .  cos  . 
2
y y
y
r r
  
r r
 

2
2
1
V
1

V
2  V
2 2 sin  .cos 
=sin  2  sin  .cos  .
r

r
r
r
2
1

cos  V  sin  .cos  . V

r
r r r
2
1

V
2
 2 cos  2
r

2
Adding the result, we get
 V  1 cos2   sin 2   V
V V
2
2
 2  cos   sin 
2
2
2
2

x y
r r
2
2



2
1
V
2
2
+  cos   sin  
r
r
 V 1  V 1 V
=
 2. 2  .
2
r
r 
r r
2
2

2
Q 8 : If u = x  y and v = 2xy and f (x,y) = (u,v)
2
2
2
2

2 f 2 f




2
2
then show that
 2  4 x  y  2  2 
2
x
y
v 
 u
Sol:
We have
u
u
 2 x and
=-2y
x
y
v
v
 2 y and = 2 x
x
y
f
 u  v
We have


x u x v x
f


 2x
 2y
x
u
v



 2x
 2y
as f (x,y) = (u,v)
x
u
v
 f
  
 
 
 2*2  x  y  x  y 
2
x
v  u
v 
 u
2
 2
 f

2   
 4 x
 2 xy
y
2
2
2 
x
uv
v 
 u
2
2
2
2
again we have
f
 u  v


y u y v y


 2 y
 2x
u
v

 
 
 2  y
x
as f (x,y) = (u,v)

y
v 
 u
 f
  
 
 
  2  2   y  x  y
x 
2
y
v  u
v 
 u
2
2
 2  2
2 f
 2


2
 4 y
 2 xy
x
2
2
2 
y
uv
v 
 u
Adding the result we get
 2
 f  f

2   
 2  4 x
 2 xy
y
2
2
2 
x
y
uv
v 
 u
2
2
 2  2

2   
 4 y
 2 xy
x
2
2 
uv
v 
 u
2
2
2
2
2
 f  f

2
2  
 2  4 x  y  2  2 
2
x
y
v 
 u
2
2
2
2
Exercise
1. If z = xm yn, then prove that
dz
dx
dy
m 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.
6. If
7. If
sin u
cos y
cos x
z
, u
,v 
find z / x
cos v
sin x
sin y
dz
1 y
t
z  tan
where x  log t , y  e , find
.
x
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
z / x
u=yex, y=xe-y,
and
z / y
w=y/x.
if z = u2+v2+w2, where
12.
13.
If
z=eax+byf(ax-by),
If
prove that
x 1  y   y 1  x   a
2
2
2
d y
a

3/2
2
2
dx
1  x 
14.
Find dy/dx if
(i) x4+y4=5a2axy.
(ii) xy+yx=(x+y)x+y
z
z
b a
 2abz.
x
y
, show that