# Partial Differentiation

### Defination of Partial Differentiation

If f is a function of several variables $x_1 , x_2 , \cdots x_n$ , then the derivative of f w.r.t. $x_1$ keeping other variables constant is called partial derivative of f w.r.t. $x_1$ and is denoted by $\dfrac{\delta f}{\delta x_1}$ or by $f_1$ and is defined as:

$\dfrac{\delta f}{\delta x_1} = \underset{h \rightarrow 0}{ lim} \dfrac{f ( X_1 + h , x_2 , x_3 , \cdots x_n ) - f ( x_1 , x_2 , \cdots x_n ) }{h}$ provided the limit exists.

$2 . \dfrac{ \delta ^2 f}{ \delta ^2 f} = \dfrac{\delta}{\delta x} ( \dfrac{\delta f}{\delta x} ) , \dfrac{\delta ^2 f}{\delta y^2} = \dfrac{\delta}{\delta y} ( \dfrac{\delta f}{\delta y} )$

$\dfrac{\delta ^2f}{\delta x \delta y} = \dfrac{\delta}{\delta x} ( \dfrac{\delta f}{\delta y} ) , \dfrac{\delta ^2 f}{\delta y \delta x} = \dfrac{\delta}{\delta y} ( \dfrac{\delta f}{\delta x} )$

3. if f=f(x,y) and partial derivates are continuous then $\dfrac{ \delta ^2 f}{\delta x \delta y} = \dfrac{\delta ^2 f}{\delta y \delta x}$

4. if f=f(x,y), then $df = ( \dfrac{\delta f}{\delta x} ) dx + ( \dfrac{ \delta f}{ \delta y} ) dy$ . if g=g (x,y,z), then $dg = ( \dfrac{\delta g}{\delta x} dx + ( \dfrac{ \delta g}{ \delta y} dy + ( \dfrac{ \delta g}{ \delta z} ) dz$ .

if f=f(x,y) and $x = \phi ( t ) , y = \psi ( t )$ , then:

$df = ( \dfrac{ \delta f}{ \delta x} ) dx + ( \dfrac{ \delta f}{ \delta y} ) dy$

And so $\dfrac{df}{dt} = ( \dfrac{ \delta f}{ \delta x} ) \dfrac{dx}{dt} + ( \dfrac{ \delta f}{\delta y} ) \dfrac{dy}{dt}$

### Homogenous Function

If f is a homogenous function of x and y of degree n, then it may be put in the form $f = x^n F ( \dfrac{y}{x} )$ .

Example:

$f = X^4 + 4 X^3 y + y^4$ $= X^4 [ 1 + 4 ( \dfrac{y}{x} ) + ( \dfrac{y}{x} ) ^4 ] = x^4F ( \dfrac{y}{x} )$

### Euler’s theorem

If f is a homogenous function of x and y of degree n , then $x \dfrac{ \delta f}{ \delta x} + y \dfrac{ \delta f}{ \delta y} = nf$

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

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

Note:

(1) if $f = \dfrac{ x^{ \dfrac{2}{5}} + x^{ \dfrac{4}{5}} , y^{\dfrac{3}{5}}}{xy^2 + x^2 y}$ , then degree of f is $\dfrac{7}{5} - 3 = - \dfrac{8}{5}$ and f is homogenous.

(2) If $u = sin ^{-1} ( \dfrac{ X^2 y}{xy^3 + X^4} )$

Then $sin u = f= \dfrac{x^2 y}{xy^3 + x^4}$ is homogenous degree 3 -4 = -1.

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