total diff
TRANSCRIPT
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Total Differentiation
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DIFFERENTIATION OF COMPOSITE FUNCTION
Let z = f(x,y)Possesses continuous partial derivatives and let
x = g (t)
Y= h(t)
Possess continuous derivatives
Then, . .dz z x z y
dt x t y t
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CHANGE OF VARIABLES
Let z=f(x,y)......................(1)
Possess continuous first order partial derivatives w.r.t. x,y.
Let x = (u,v) and y = (u,v)
Possesses continuous first order partial derivatives. We write,
=zu
. . .............................(2)
= . . .............................(3)
Now, solve equation (2) and (3) as simultaneously linear equations
in and . This wil
z x z yx u y u
z z x z y
v x v y v
z z
x y
l give us expressions for and in terms
of , and the easily determined quantities , , , .
z z
x y
z z x x y y
u v u v u v
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Differentiation of Implicit Function
Let f(x,y) = 0 defines y as a function of x implicitly.
We shall obtain the value of in terms of the partial derivatives
and .
Since f(x,y) is a function of x and y and y is function of x,
dy
dx
f f
x y
therefore
we can look upon f(x,y) as a composite function of x.
f f= . .
f f0 . ...........................( )
-
df dx dy
dx x dx y dx
dyi
x y dx
dy f x
dx f y
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1
2 2
2 2
yExample 1: If x = tan , prove that
x
x dy - y dxdz = x + y
Solution: We know that,
dz = . .
1But, .
1
z zdx dyx y
z y
y xy x
2 2=
x + y
y
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2
2 2
2 2 2 2
2 2
1and =
1
x + y
dz =x + y x + y
xdy - ydx=x + y
z x
y y x
x
y xdx dy
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2 2 2
2 2
2 2
2: Find dz/dt when
z=xy , x=at , y = 2at
Verify by direct substitution.
Solution. We have
z= xy
2 and 2
2 and 2
= . .
Example
x y
x y
z zy xy x xy
x y
dx dyat adt dt
dz z dx z dy
dt x dt y dt
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2 2
2 2 2 3 2 3 2 4
3 3 4
2 2
3 5 3 4 3 3 4
( 2 ).2 (2 ).2
(4 4 ).2 (4 ).2
(16 10 )
,
2 4 (16 10 )
Hence the verification
y xy at xy x a
a t a t at a t a t a
a t t
Again
z x y xy
a t a t a t t
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2 2
2
Example 3: If z=x and y=z , then find the differential co-
efficient of the first order when x is the independent variable.
Solution: dz=
Since z=x 2 , 1
Thus, dz=2xdx+dx+2z
y x
z zdx dyx y
z zy x
x y
dz
dz(1-2z) = dx(2x+1)
dz (2x+1)
dx (1-2z), 2 (2 ) (1 4 ) 2
(1 2 ) (1 4 )
(1 4 )
(1 2 )
Also dy dx z xdx dy dx xz zdy
dy z dx xz
dy xz
dx z
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Example 4: z is a function of x and y, prove that if x = eu + e-v,
y = e-u + e-v then
z z z zx y
du dv dx dy
Solution: z is a composite function of u, v. we havez z z z
x ydu dv dx dy
. .
u uz ze y e
dx dy
. .
= - . .v v
z z x z ydv dx dv dy dv
z ze e
dx dy
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Subtracting, we get
= x
u v u vz z z ze e e edu dv dx dy
z z
ydx dy
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Example 5: If z = ex sin y, where x = In t and y = t2, then finddz
dtSolution: We know that,
. .dz z dx z dy
dt x dt y dt
But ,sin ,
xze y
x
2
cos ,
1and 2
1sin . ( cos )2
= (sin 2 cos )
x
x x
x
ze y
y
dx dyt
dt t dt
dze y e y t
dt t
ey t y
t
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Example 6: If H = f(y-z, z-x, x-y), prove that
0H H H
x y z
Solution: Let, u = y-z, v = z-x, w = x-y
H = f(u,v,w)H is a composite function of x,y,z.
We have,
. . .
= .0 .( 1) .1
=-
H H u H v H wx u x v x w x
H H H
u v w
H Hv w
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Similarly,H H H
y w uH H H
z u v
Adding all the above, we get
0H H H
x y z
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Example 7: If x = r cos, y = r sin and V=f(x,y), then show that
2 2 2 2
2 2 2 2 2
1 1.
V V V V V
x y r r r r
Solution: We have, x = r cos, y = r sin
cos . sin ..................( )
dy=sin .dx+sin . ...............( )
dx dr r i
and dy ii
Solving (i) and (ii) as simultaneous linear equations in dr and d, we getDr = cos.dx + sin. dy (iii)
1(cos . sin . ).....................( )
, dr = ....................( )
.....................( )
d dy dx ivr
r rNow dx dy vx y
and d dx dy vix y
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Comparing (iii) and (v), we getcos , sin
r r
x y
Comparing (iv) and (vi), we get
1 1sin , cos
x r y r
Thus,
. .
1=cos . sin
1= cos . sin ..................( )
v v r v
x r x x
v vr r
v viir r
Similarly, . .
1=cos . sin
1= sin . cos ..................( )
v v r vy r y y
v v
r r
v viiir r
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It follows that from (vii)2
2
2 2 2 22
2
22
2 2 2
1 1cos . sin . cos . sin .
1 1 sin=cos sin .cos . sin .cos .
1 1sin sin .cos ...................( )
V v v
x r r r r
V V V V
r r r r r r r
V Vix
r r
Similarly from (viii) it follows that2
2
2 2 2 22
2
22
2 2 2
1 1sin . cos . sin . cos .
1 1 cos=sin sin .cos . sin .cos .
1 1cos sin .cos ...................( )
V v v
y r r r r
V V V V
r r r r r r r
V Vx
r r
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Adding (ix) and (x), we get
2 2 2 2
2 2 2 2
2 2 2 2 2
2 2
2 2
2 2 2
1
cos sin cos sin
1+ cos sin
1 1= . .
V V V V
x y r r
V
r r
V V V
r r r r
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1. If z = xm yn, then prove that
dz dx dym nz x y
2. If u = x2-y2, x=2r-3s+4, y=-r+8s-5, find /u r 3. If x=r cos, y=r sin, then show that
(i) dx = cos.dr-r sin.d(ii) dy = sin.dr+r.cos.dDeduce that
(i)dx2+dy2=dr2+r2d2(ii)xdy-ydx=r2.d4. If z=(cosy)/x and x=u2-v, y=eV, find /z v 5. If z=x2+y and y=z2+x, find differential co-efficients of the first
order when(i)y is the independent variable.(ii)z is the independent variable.
Exercise
i
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6. If
sin cos cos, , /
cos sin sin
u y xz u v find z x
v x y
7. If
1
tan log , , .ty dz
z where x t y e findx dt
8. If u = (x+y)/(1-xy), x=tan(2r-s2), y=cot(r2s) then find9. If z=x2-y2, where x=etcost, y=etsint, find dz/dt.10. If z=xyf(x,y) and z is constant, show that
'( / ) [ ( / )]
( / ) [ ( / )]
f y x x y x dy dx
f y x y y x dy dx
11. Find and if z=u2+v2+w2, where u=yex,y=xe-y, w=y/x./z x /z y
f z f
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12. If f(x,y)=0, (y,z)=0, show that . . .f z f
y z x x y
13. If z=eax+byf(ax-by), prove that2 .
z z
b a abzx y
14. If 2 21 1x y y x a , show that
2
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1
d y a
dx x
15. Find dy/dx if
(i) x4+y4=5a2axy.(ii) xy+yx=(x+y)x+y