PARTIAL DIFFERENTIATION 345

G

t= 2 π

√

l

g

=

(

2 π

√

g

)√

l=

(

2 π

√

g

)

l

1

2

Hence

∂t

∂l

=

(

2 π

√

g

)

d

dl

(l

1

(^2) )=
(
2 π
√
g
)(
1
2
l
− 1
2
)

(
2 π
√
g
)(
1
2
√
l
)

π
√
lg
To find
∂t
∂g
,lis kept constant
t= 2 π
√
l
g
=(2π
√
l)
(
1
√
g
)
=(2π
√
l)g
− 1
2
Hence
∂t
∂g
=(2π
√
l)
(
−
1
2
g
− 3
2
)
=(2π
√
l)
(
− 1
2
√
g^3
)

−π
√
l
√
g^3
=−π
√
l
g^3
Now try the following exercise.
Exercise 139 Further problems on first
order partial derivatives
In Problems 1 to 6, find
∂z
∂x
and
∂z
∂y
1.z= 2 xy
[
∂z
∂x
= 2 y
∂z
∂y
= 2 x
]
2.z=x^3 − 2 xy+y^2
⎡
⎢
⎣
∂z
∂x
= 3 x^2 − 2 y
∂z
∂y
=− 2 x+ 2 y
⎤
⎥
⎦
3.z=
x
y
⎡
⎢
⎣
∂z
∂x

1
y
∂z
∂y

−x
y^2
⎤
⎥
⎦
4.z=sin (4x+ 3 y)
⎡
⎢
⎣
∂z
∂x
=4 cos (4x+ 3 y)
∂z
∂y
=3 cos (4x+ 3 y)
⎤
⎥
⎦
5.z=x^3 y^2 −
y
x^2

• 1 y ⎡ ⎢ ⎢ ⎣
  ∂z
  ∂x
  = 3 x^2 y^2 +
  2 y
  x^3
  ∂z
  ∂y
  = 2 x^3 y−
  1
  x^2
  −
  1
  y^2
  ⎤
  ⎥
  ⎥
  ⎦
  6.z=cos 3xsin 4y
  ⎡
  ⎢
  ⎢
  ⎣
  ∂z
  ∂x
  =−3 sin 3xsin 4y
  ∂z
  ∂y
  =4 cos 3xcos 4y
  ⎤
  ⎥
  ⎥
  ⎦

7. The volume of a cone of heighthand base
   radiusris given byV=^13 πr^2 h. Determine
   ∂V
   ∂h

and

∂V

∂r [

∂V

∂h

=

1

3

πr^2

∂V

∂r

=

2

3

πrh

]

8. The resonant frequencyfrin a series electrical

circuit is given byfr=

1

2 π

√

LC

. Show that

∂fr

∂L

=

− 1

4 π

√

CL^3

9. An equation resulting from plucking a
   string is:

y=sin

(nπ

L

)

x

{

kcos

(

nπb

L

)

t+csin

(

nπb

L

)

t

}

Determine

∂y

∂t

and

∂y

∂x

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂y

∂t

=

nπb

L

sin

(nπ

L

)

x

{

ccos

(

nπb

L

)

t

−ksin

(

nπb

L

)

t

}

∂y

∂x

=

nπ

L

cos

(nπ

L

)

x

{

kcos

(

nπb

L

)

t

+csin

(

nπb

L

)

t

}

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

10. In a thermodynamic system,k=Ae

TS−H

RT ,

whereR,kandAare constants.

Find (a)

∂k

∂T

(b)

∂A

∂T

(c)

∂(S)

∂T

(d)

∂(H)

∂T