Partial differentiation 347
To find
∂t
∂l
,gis kept constant.
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
=( 2 π
√
l)
(
1
√
g
)
=( 2 π
√
l)g
− 1
2
Hence
∂t
∂g
=( 2 π
√
l)
(
−
1
2
g
− 3
2
)
=( 2 π
√
l)
(
− 1
2
√
g^3
)
−π
√
l
√
g^3
=−π
√
l
g^3
Now try the following exercise
Exercise 138 Further problemson first
order partial derivatives
In Problems 1 to 6, find
∂z
∂x
and
∂z
∂y
- z= 2 xy
[
∂z
∂x
= 2 y
∂z
∂y
= 2 x
]
- z=x^3 − 2 xy+y^2
⎡
⎢
⎢
⎣
∂z
∂x
= 3 x^2 − 2 y
∂z
∂y
=− 2 x+ 2 y
⎤
⎥
⎥
⎦
- z=
x
y
⎡
⎢
⎣
∂z
∂x
=
1
y
∂z
∂y
=
−x
y^2
⎤
⎥
⎦
- z=sin( 4 x+ 3 y)⎡
⎢
⎢
⎣
∂z
∂x
=4cos( 4 x+ 3 y)
∂z
∂y
=3cos( 4 x+ 3 y)
⎤
⎥
⎥
⎦
- 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
⎤
⎥
⎥
⎦
- z=cos3xsin4y
⎡
⎢
⎢
⎣
∂z
∂x
=−3sin3xsin4y
∂z
∂y
=4cos3xcos4y
⎤
⎥
⎥
⎦
- 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
]
- The resonant frequencyfrin a series electri-
cal circuit is given byfr=
1
2 π
√
LC
.Show
that
∂fr
∂L
=
− 1
4 π
√
CL^3
- 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
}
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦