…………….

1. If where

Prove that

2.     If V = show that

3.     If show that

4.     If then prove that

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)   

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

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

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

1. If be a homogeneous function of degree n in x & y then,

.

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

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.

1.   Calculate
2. If
3. If

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.

1. If

Where

2.     If   and

Find

3.     If

Find

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