Partial differentiation 347
To find
∂t
∂l,gis kept constant.t= 2 π√
l
g=(
2 π
√
g)
√
l=(
2 π
√
g)
l1
2Hence
∂t
∂l=(
2 π
√
g)
d
dl(l1(^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
3πr^2∂V
∂r=2
3πrh]- The resonant frequencyfrin a series electri-
cal circuit is given byfr=
1
2 π√
LC.Showthat∂fr
∂L=− 1
4 π√
CL^3- An equation resulting from plucking a
string is:
y=sin
(nπL)
x{
kcos(
nπb
L)
t+csin(
nπb
L)
t}Determine∂y
∂tand∂y
∂x
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
∂y
∂t=nπb
Lsin(nπ
L)
x{
ccos(
nπb
L)
t−ksin(
nπb
L)
t}∂y
∂x=nπ
Lcos(nπ
L)
x{
kcos(
nπb
L)
t+csin(
nπb
L)
t}⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦