Application Of Partial derivative for F Y.docx

Partial Derivative
Let us consider a function u of three independent variable x,y,z i.e.
u=f (x , y , z)−−−−(1)
Then the partial derivative of u w.r.t x is denoted by
∂u
∂x
and is defined as
∂u(x, y , z)
∂x
= lim
∆ x→0
f (x+∆ x ,z)−f (x , y , z)
∆ x
The partial derivative of u w.r.t. y and z can be defined similarly and they are
denoted by
∂u
∂ y
&
∂u
∂ z
.
The partial derivative
∂u
∂x
is denoted by as ux
Similarly
∂u
∂ y
=uy &
∂u
∂ z
=uz.
The partial derivative of higher order of the function (1) can also be calculated by
successive differentiation. Thus we have
∂
2
u
∂x
2
=
∂
∂ x (∂u
∂ x )=uxx
∂
2
u
∂ y
2
=
∂
∂ y (∂u
∂ y )=uyy
∂2
u
∂x ∂ y
=
∂
∂x (∂u
∂ y )=uxy
∂
2
u
∂ y∂ x
=
∂
∂ y (∂u
∂ x )=uyx
Problems:
1. If u=x3
−3 xy2
then prove that
∂
2
u
∂x
2
+
∂
2
u
∂ y
2
=0
2. If u=eax
sinby verify
∂2
u
∂x ∂ y
=
∂2
u
∂ y ∂ x
3. If z=log ⁡(x2
+ y2
) then prove that
∂
2
z
∂x ∂ y
=
∂
2
z
∂ y ∂ x
Mrs P M Swami, Department of BSH

4. If u=log ⁡(x3
+ y3
+z3
−3xyz) prove that
∂u
∂x
+
∂u
∂ y
+
∂u
∂ z
=
3
x+ y+z
5. Ifu=exyz
find
∂3
u
∂x ∂ y ∂z
6. If u=e
x
2
+ y
2
+z
2
then prove that
∂
3
u
∂x ∂ y ∂z
=8 xyzu
7. Ifz (x+ y)=x2
+ y2
then prove that (
∂ z
∂ x
−
∂z
∂ y
)
2
=4[1−
∂ z
∂ x
−
∂ z
∂ y ]
8. If u=x
y
verify
∂2
u
∂x ∂ y
=
∂2
u
∂ y ∂ x
9. Prove that z=f (x+at)+∅(x −at) satisfies
∂
2
z
∂t
2
=a
2 ∂
2
z
∂ x
2 where a is constant.
10. If z=x2
tan− 1 y
x
− y2
tan− 1 x
y
prove that
∂
2
z
∂x ∂ y
=
x
2
− y
2
x2
+ y2
11.If u=log ⁡(x
3
+ y
3
−x
2
y − y
2
x) prove that (
∂
∂ x
+
∂
∂ y
)
2
u=−
4
(x+ y)
2
12.If u=log ⁡(x
3
+ y
3
−z
3
+3xyz) prove (
∂
∂ x
+
∂
∂ y
+
∂
∂z
)
2
u=−
9
(x+ y+z)2
13.If u=
1
x
2
+ y
2
+ z
2 then prove that
∂2
u
∂x2
+
∂2
u
∂ y2
+
∂2
u
∂z2
=2u
2
14. If r
2
=x
2
+ y
2
+z
2
∧V=r
m
, provet at
ℎ V xx+V yy +Vzz =m(m+1)r
m−2
15.If x
x
y
y
z
z
=c , show that at x=y=z,
∂
2
z
∂x ∂ y
=−(xlogex)−1
16.Find the value of “n” so that V=r
n
(3cos
2
θ−1) Satisfies the equation
∂
∂r (r
2 ∂V
∂r )+
1
sinθ
∂
∂θ (sinθ
∂V
∂θ )=0

Examples on variable to be treated as constant:
(
∂r
∂ x
)
θ
Means the partial derivative of r w.r.t x treating θ as constant. In a relation
expressing r is function of x and θ only.
When no indication is given regarding which variable to be kept constant then by
convention we take
∂
∂x
→
(
∂
∂ x
)
y
∧∂
∂ y
→(
∂
∂ y
)
x
Similarly ∂
∂r
→
(
∂
∂r
)
θ
∧∂
∂θ
→(
∂
∂θ
)
r
Problems:
1. If x=rcosθ∧y=rsinθ then ST (i) [x(∂ x
∂r )θ
+ y(∂ y
∂r )θ
]
2
=x2
+ y2
(ii) (
∂ y
∂θ
)
r
(iii) (
∂r
∂ x
)
y
(iv) (
∂θ
∂ y
)
x
(iv) (
∂θ
∂ x
)
y
Differentiation of composite functions:
Let z=f (x , y),x=∅(t )∧ y=∅(t) so that z is function of x,y and x & y are themselves
function of third variable t. these three relation defines z is function of t.
therefore z is called composite function of t.
z→ x , y →t
z→t
dz
dt
=
∂ z
∂x
.
dx
dt
+
∂ z
∂ y
.
dy
dt
If z is function of x & y and z=f (x , y),x=∅(u,v)∧ y=∅(u,v) then
z→ x , y →u, v
That is z→u,v
∂ z
∂u
=
∂ z
∂x
.
∂x
∂u
+
∂ y
∂u
.
∂ z
∂ y

∂ z
∂v
=
∂ z
∂x
.
∂ x
∂v
+
∂ y
∂v
.
∂ z
∂ y
Problems:
1. If z=x
2
+ y
2
,x=atsint∧y=atcostthen find
dz
dt
.
2. If z=tan
−1 x
y
,x=2t∧y=1−t
2
find
dz
dt
.
3. If z=x
2
+ y
2
,x=at
2
∧y=2at find
dz
dt
.
4. If u=xlogxy ,w ere
ℎ x
3
+ y
3
+3xy=1,find
du
dx
5. If u=f (ex − y
,ey− z
,ez−x
) Find
∂u
∂x
+
∂u
∂ y
+
∂u
∂ z
6. If u=f (ax −by ,by−cz ,cz−ax)Find
1
a
∂u
∂x
+
1
b
∂u
∂ y
+
1
c
∂u
∂z
7. If u=f (x− y , y −z ,z −x) then Find
∂u
∂x
+
∂u
∂ y
+
∂u
∂ z
8. If u=f (x2
− y2
, y2
−z2
,z2
−x2
) then Find
1
x
∂u
∂x
+
1
y
∂u
∂ y
+
1
z
∂u
∂ z
Differentiation of implicit function of two variable:
dy
dx
=−
∂f
∂ x
∂f
∂ y
Problems:
1. If (cosx)
y
=(siny)
x
find
dy
dx
2. If x
y
+ y
x
+x
x
+ y
y
=0 find
dy
dx

Homogenous function:
A function f(x,y,z) is called homogenous function of degree n. if by putting
X=xt, Y=yt and Z=zt the function of f(x,y,z) becomes tn
f(x,y,z). where n is
degree.
Problems:
1. Verify whether the function u=
x
2
+ y
2
√x+√ y
is homogenous or not.
2. Verify whether the function u=sin(x
2
+ y
2
x+ y ) is homogenous or not.
3. Verify whether the function u=sin
−1 y
x
is homogenous or not.
Euler’s theorem:
Statement: For a homogenous function u(x,y) of degree n in x & y is
x
∂u
∂ x
+ y
∂u
∂ y
=nu
Proof:
Since u(x,y) is homogenous function of degree n in x & y hence it will be
expressed as u=xn
∅(x
y )−−−−−−−(1)
∂u
∂x
=x
n
∅
'
(x
y ).
1
y
+∅(x
y )nx
n−1
x
∂u
∂ x
=xn+1
∅'
(x
y ).
1
y
+∅(x
y )n xn
x
∂u
∂ x
=x
n+1
∅
'
(x
y ).
1
y
+nu−−−−−−(2)
∂u
∂ y
=x
n
∅
'
(x
y )(−
x
y
2 )
y
∂u
∂ y
=−xn+1
∅'
(x
y ).
1
y
−−−−−(3)

From (2) & (3) we have x
∂u
∂ x
+ y
∂u
∂ y
=nu
Theorem: If u is homogenous function of degree n in x & y then prove that
x2 ∂2
u
∂ x2
+2xy
∂2
u
∂ x∂ y
+ y2 ∂2
u
∂ y2
=n(n−1)u
Proof: we know that x
∂u
∂ x
+ y
∂u
∂ y
=nu−−−−−(1)
Differentiate w.r.t x partially,
x
∂
2
u
∂ x
2
+
∂u
∂x
+ y
∂
∂ x (∂u
∂ y )=n
∂u
∂ x
x
2 ∂2
u
∂ x2
+x
∂u
∂ x
+xy
∂2
u
∂ x ∂ y
=nx
∂u
∂x
x
2 ∂
2
u
∂ x
2
+xy
∂
2
u
∂ x ∂ y
=(n−1)x
∂u
∂ x
−−−−−(2)
Differentiate w.r.t y partially,
x
∂
∂ y (∂u
∂x )+ y
∂
2
u
∂ y
2
+
∂u
∂ y
=n
∂u
∂ y
xy
∂2
u
∂ y ∂ x
+ y2 ∂2
u
∂ y
2
+ y
∂u
∂ y
=ny
∂u
∂ y
xy
∂2
u
∂ y ∂ x
+ y
2 ∂2
u
∂ y2
=(n−1) y
∂u
∂ y
−−−−−(3)
From (2) & (3) we have
x
2 ∂
2
u
∂ x
2
+2xy
∂
2
u
∂ x∂ y
+ y
2 ∂
2
u
∂ y
2
=n(n−1)u
Corollary: If u is homogenous function of three variable x,y & z of degree n then

x
∂u
∂ x
+ y
∂u
∂ y
+z
∂u
∂ z
=nu
Corollary: If z is homogenous function of x & y and z=f(u) then,
x
∂u
∂ x
+ y
∂u
∂ y
=n
f (u)
f '
(u)
And x
2 ∂
2
u
∂ x2
+2xy
∂
2
u
∂ x∂ y
+ y
2 ∂
2
u
∂ y2
=g(u)[ g
'
(u)−1]
Where g(u)=n
f (u)
f '
(u)
Problems:
1. If z=
x
1
3
+ y
1
3
x
1
2
+ y
1
2
find x
∂z
∂ x
+ y
∂z
∂ y
2. If u=
x3
y+ y3
x
y− x
then find x
2 ∂
2
u
∂ x2
+2xy
∂
2
u
∂ x∂ y
+ y
2 ∂
2
u
∂ y2
3. If u=xsin
− 1 y
x
+ y tan
−1 y
x
then find
i. x
∂u
∂ x
+ y
∂u
∂ y
ii. x
2 ∂2
u
∂ x2
+2xy
∂2
u
∂ x∂ y
+ y
2 ∂2
u
∂ y2
4. If u=x
4
y
2
tan
−1 x
y
find
i. x
∂u
∂ x
+ y
∂u
∂ y
ii. x2 ∂
2
u
∂ x2
+2xy
∂
2
u
∂ x∂ y
+ y2 ∂
2
u
∂ y2
5. If u=sin
−1
(√x−√y
√x+√ y )t en
ℎ provet at
ℎ
∂u
∂ y
=−
x
y
∂u
∂ x
6. If y=xcosu t en
ℎ find x
2 ∂
2
u
∂ x
2
+2 xy
∂
2
u
∂ x∂ y
+ y
2 ∂
2
u
∂ y
2

7. If u=f (x
y
+
y
x
+
z
x )t en
ℎ find x
∂u
∂ x
+ y
∂u
∂ y
8. Verify Euler’s theorem for the function u=log ⁡(
x+ y
x− y
)
9. If u=4 log
(x
1
4
+ y
1
4
x
1
5
+ y
1
5 )then find x
2 ∂2
u
∂ x2
+2xy
∂2
u
∂ x∂ y
+ y
2 ∂2
u
∂ y2
10. If u=sin−1
√x2
+ y2
then PT x
2 ∂2
u
∂ x
2
+2xy
∂2
u
∂ x∂ y
+ y
2 ∂2
u
∂ y
2
=tan
3
u
11.If u=cosec
−1
√x
1
2
+ y
1
2
x
1
3
+ y
1
3
then prove that
x
2 ∂2
u
∂ x
2
+2xy
∂2
u
∂ x∂ y
+ y
2 ∂2
u
∂ y
2
=
tanu
12 [13
12
+
tan2
u
12 ]
12.Verify Euler’s theorem for the function u=sin−1
√x2
+ y2
13.If cosu=
√x2
+ y2
x− y
then find x
2 ∂
2
u
∂ x2
+2xy
∂
2
u
∂ x∂ y
+ y
2 ∂
2
u
∂ y2

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