346 Higher Engineering Mathematics
(b) To find
∂z
∂y
,xis kept constant.
Sincez=( 5 x^4 )+( 2 x^3 )y^2 − 3 y
then,
∂z
∂y
=( 5 x^4 )
d
dy
( 1 )+( 2 x^3 )
d
dy
(y^2 )− 3
d
dy
(y)
= 0 +( 2 x^3 )( 2 y)− 3
Hence
∂z
∂y
= 4 x^3 y− 3.
Problem 2. Giveny=4sin3xcos2t,find
∂y
∂x
and
∂y
∂t
To find
∂y
∂x
,tis kept constant.
Hence
∂y
∂x
=(4cos2t)
d
dx
(sin3x)
=(4cos2t)(3cos3x)
i.e.
∂y
∂x
=12cos3xcos2t
To find
∂y
∂t
,xis kept constant.
Hence
∂y
∂t
=(4sin3x)
d
dt
(cos 2t)
=(4sin3x)(−2sin2t)
i.e.
∂y
∂t
=−8sin3xsin2t
Problem 3. Ifz=sinxyshow that
1
y
∂z
∂x
=
1
x
∂z
∂y
∂z
∂x
=ycosxy,sinceyis kept constant.
∂z
∂y
=xcosxy,sincexis kept constant.
1
y
∂z
∂x
=
(
1
y
)
(ycosxy)=cosxy
and
1
x
∂z
∂y
=
(
1
x
)
(xcosxy)=cosxy.
Hence
1
y
∂z
∂x
=
1
x
∂z
∂y
Problem 4. Determine
∂z
∂x
and
∂z
∂y
when
z=
1
√
(x^2 +y^2 )
z=
1
√
(x^2 +y^2 )
=(x^2 +y^2 )
− 1
2
∂z
∂x
=−
1
2
(x^2 +y^2 )
− 3
(^2) ( 2 x),by the function of a
function rule (keepingyconstant)
−x
(x^2 +y^2 )
3
2
−x
√
(x^2 +y^2 )^3
∂z
∂y
=−
1
2
(x^2 +y^2 )
− 3
(^2) ( 2 y),(keepingxconstant)
−y
√
(x^2 +y^2 )^3
Problem 5. Pressurepof a mass of gas is given
bypV=mRT,wheremandRare constants,Vis
the volume andTthe temperature. Find expressions
for
∂p
∂T
and
∂p
∂V
.
SincepV=mRTthenp=
mRT
V
To find
∂p
∂T
,Vis kept constant.
Hence
∂p
∂T
(
mR
V
)
d
dT
(T)=
mR
V
To find
∂p
∂V
,Tis kept constant.
Hence
∂p
∂V
=(mRT)
d
dV
(
1
V
)
=(mRT)(−V−^2 )=
−mRT
V^2
Problem 6. The time of oscillation,t,of
a pendulum is given byt= 2 π
√
l
g
wherelis the
length of the pendulum andgthe free fall
acceleration due to gravity. Determine
∂t
∂l
and
∂t
∂g