Unit –3
Partial Differentiation
Partial Differentiation
If
Prove that
Partial differentiation of function of function
If z = f(u) and . Then z becomes a function of x & y. In this case z becomes a function of function of x & y.
i.e.

Then
,
Similarly
If
Then z becomes a function of function of x, y & z.
…………….
Prove that 2. If V = show that 3. If show that 4. If then prove that

Partial Differentiation of composite function
a) Let and , then z becomes a function of , In this case z is called composite function of . i.e. b) Let possess continuous partial derivatives and let possess continuous partial derivatives, then z is called composite function of x and y. i.e.
& Continuing in this way, ….. Ex. If Then prove that
Ex. If then prove that Where is function of x, y, z.
Ex. If where , then show that, i) ii)

Notations of partial derivatives of variable to be treated as a constant
Let and i.e. Then means partial derivative of u w.r.t. x treating y const. To find from given reactions we first express x in terms of u & v. i.e. & then diff. x w.r.t. u treating v constant. To find express v as a function of y and u i.e. then diff. v w.r.t. y treating u as a const.
Ex. If , then find the value of . Ex. If , then prove that

A polynomial in x & y is said to be Homogeneous expression in x & y of degree n. if the degree of each term in the expression is same & equal to n.
e.g.
is a homogeneous function of degree 3.
To find the degree of homogeneous expression f(x, y).
 Consider
 Put . Then if we get .
Then the degree of is n.
Ex.
Consider 
Put 
. 
Thus degree of f(x, y) is 
Note that 
If be a homogeneous function of degree n then z can be written as 
Differentiation of Implicit function
Suppose that we cannot find y explicitly as a function of x. but only implicitly through the relation f(x, y) = 0. Then we find Since diff. P. w.r.t. x we get i.e. Similarly, it f (x, y, z) = 0 then z is called implicit function of x, y. then in this case we get
Ex. Find if
Ex. Find . If , & Ex. If , where Find
Ex. If Then find Eulers Theorem on Homogeneous functions: Statement: If be a homogeneous function of degree n in x & y then,
Deductions from Eulers theorem
. 2. If be a homogeneous functions of degree n in x & y and also then, And Where
Ex. If , find the value of
Ex. If then find the value of
Ex. If then prove That
Ex. If the prove that
Ex. If then show That

Jacobians, Errors and Approximations, maxima and minima
Jacobians If u and v be continuous and differentiable functions of two other independent variables x and y such as , then we define the determine as Jacobian of u, v with respect to x, y Similarly , JJ’ = 1 Actually Jacobins are functional determines Ex.
ST 4. find 5. If and , find 6. 7. If 8. If , , JJ1 = 1 If , JJ1=1 Jacobian of composite function (chain rule) Then Ex.
Where 2. If and Find 3. If Find

Jacobian of Implicit function
Let u1, u2 be implicit functions of x1, x2 connected by f1, f2 such there 
, 
Then 
Similarly, 
Ex. If If Find
Partial derivative of implicit functions Consider four variables u, v, x, y related by implicit function. , Then Ex. If and Find If and Find Find
If Find

Reference Books:
1. Advanced Engineering Mathematics by Erwin Kreyszig (Wiley Eastern Ltd.)
2. Advanced Engineering Mathematics by M. D. Greenberg (Pearson Education)
3. Advanced Engineering Mathematics by Peter V. O’Neil (Thomson Learning)
4. Thomas’ Calculus by George B. Thomas, (AddisonWesley, Pearson)
5. Applied Mathematics (Vol. I & Vol. II) by P.N.Wartikar and J.N.Wartikar Vidyarthi Griha Prakashan, Pune.
6. Linear Algebra –An Introduction, Ron Larson, David C. Falvo (Cenage Learning, Indian edition)