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 )− 3d
dy(y)= 0 +( 2 x^3 )( 2 y)− 3Hence∂z
∂y= 4 x^3 y− 3.Problem 2. Giveny=4sin3xcos2t,find
∂y
∂x
and∂y
∂tTo find∂y
∂x,tis kept constant.Hence∂y
∂x=(4cos2t)d
dx(sin3x)=(4cos2t)(3cos3x)i.e.∂y
∂x=12cos3xcos2tTo find∂y
∂t,xis kept constant.Hence∂y
∂t=(4sin3x)d
dt(cos 2t)=(4sin3x)(−2sin2t)
i.e.∂y
∂t=−8sin3xsin2tProblem 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)=cosxyand1
x∂z
∂y=(
1
x)
(xcosxy)=cosxy.Hence1
y∂z
∂x=1
x∂z
∂yProblem 4. Determine∂z
∂xand∂z
∂ywhenz=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