Transcript File
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
uv
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
uv
v
u
Adding the result we get
2
f f
2
2 4 x
2 xy
y
2
2
2
x
y
uv
v
u
2
2
2 2
2
4 y
2 xy
x
2
2
uv
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