PPT on partial differentiation..helpful ppts partial derivatives by Jeremy Jeremy and Ravi K Desai to make this one!!

1. L.D COLLEGE OF ENGINEERING
RUBBER TECHNOLOGY

2. ROLL NO. NAME
 126001 ADHVARYU UTKARSH
 126002 RAMOLIYA NAVDEEP
 126003 BARAIYA GOPAl
 126004 ITALIYA SHANKESH
 126005 KANETIYA JAYDEEP
 126006 LAKUM BHAVESH
 126007 AMIPARA KEYUR
 126008 SHAH ARTH
 126010 SURANI PUNAL

3.  Contents
 Function of two variables
 Limits and continuity
 Partial derivatives in first order
 Partial derivatives in higher order
PARTIAL DIFFERENTIATION

5. WHAT IS PARTIAL DIFFERENTIATION ?
Let z=f(x,y) be function of two individual variables
x & y the derivative with respect to x keeping y
constant Is called partial derivative of z with
respect to x.
It is denoted by ∂z , ∂f , fx .
∂x ∂x
It is denoted as
∂z = lim f(x+∂x ,y) – f(x,y) .
∂x ∂x→0 ∂x

8. EXAMPLE :
By considering differnent paths of approach, show that the function
f(x,y) = x4 – y2 has no limit as (x,y)→(0,0).
x4 + y2
Solution :
lim lim x4 – y2 = lim x4 = lim 1 = 1
x→0 y→0 x4 + y2 x→0 x4 x→0
lim lim x4 – y2 = lim (–y2) = lim (-1) = -1
y→0 x→0 x4 + y2 y→0 y2 x→0
Since both the limits are different, f(x,y) has no limit as (x,y)→(0,0).

11. EXAMPLE :
If a2x2+ b2y 2 = c2z2 , evaluate 1 ∂2z + 1 ∂2z .
a2 ∂x2 b2 ∂y2
Solution :
a2x2+ b2y2 = c2z2
Differentiating partially w.r.t. x,
2a2x = 2c2z ∂z
∂x
∂z = a2x
∂x c2z
Differentiating ∂z partially w.r.t. x,
∂x
∂2z = a2 (1 - x ∂z)
∂x2 c2z z2 ∂x
= a2 (1 – x a2x)
c2z z c2z

15. EXAMPLE :
If u = x3y3z3 then prove that x ∂u +y ∂u +z ∂u =6u
x3+y3+z3 ∂x ∂y ∂z
Solution :
Replacing x by xt and y by yt,
u = t6 x3y3z3
x3+y3+z3
Hence,u is a homogeneous function of degree 6.
By euler’s theorem,
x ∂u +y ∂u +z ∂u =6u
∂x ∂y ∂z

18. EXAMPLE :
If u = y2-4ax,x = at2,y = 2at,find du .
dt
Solution :
du = ∂u .dx + ∂u .dy
dt ∂x dt ∂y dt
=(-4a)2at+2(2at)(2a)
du =-8a2t+2(2at)(2a)
dt
=-8a2t+8a2t
=0

23. EXAMPLE :
If x3+y3 = 3axy,find dy .
dx
Solution :
Let f(x,y)= x3+y3-3axy
dy = ∂u/ ∂x
dx ∂f/ ∂y
= 3x2-3ay
3y2-3ax
= x2- ay
y2-ax
= ay-x2
y2-ax

24. YOU
ROLL NO. 1 TO 10