# Partial Differentiation

**Defination of Partial Differentiation**

If f is a function of several variables , then the derivative of f w.r.t. keeping other variables constant is called** partial derivative** of f w.r.t. and is denoted by or by and is defined as:

provided the limit exists.

3. if f=f(x,y) and partial derivates are continuous then

4. if f=f(x,y), then . if g=g (x,y,z), then .

if f=f(x,y) and , then:

And so

**Homogenous Function**

If f is a homogenous function of x and y of degree n, then it may be put in the form .

Example:

**Eulerâ€™s theorem**

If f is a *homogenous function* of x and y of degree n , then

**Deduction**: From this, we get some important results as follows:

**Note**: The above results are very important for doing problems.

**Note**:

(1) if , then degree of f is and f is homogenous.

(2) If

Then is **homogenous degree** 3 -4 = -1.

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