Partial Differentiation
The partial differential coefficient of f ( x , y ) with respect to x is the ordinary differential coefficient of f ( x , y ) when y is regarded as a constant . It is a written as ∂f / ∂x or Dxf or fx .
e . g ., If z = f ( x , y ) = x 4 + y 4 + 3xy 2 + x 4 y + x + 2y
Then , ∂z / ∂x or ∂f / ∂x or fx = 4x 3 + 3y 2 + 2xy + 1 ( here , y is consider as constant )
∂z / ∂y or ∂f / ∂y or fy = 4y 3 + 6xy + x 2 + 2 ( here , x is consider as constant ) Higher Partial Derivatives Let f ( x , y ) be a function of two variables such that ∂f / ∂x , ∂f / ∂y both exist . ( i ) The partial derivative of ∂f / ∂y w . r . t . ‘ x ’ is denoted by ∂ 2 f / ∂x 2 / or fxx . ( ii ) The partial derivative of ∂f / ∂y w . r . t . ‘ y ’ is denoted by ∂ 2 f / ∂y 2 / or fyy .
( iii ) The partial derivative of ∂f / ∂x w . r . t . ‘ y ’ is denoted by ∂ 2 f / ∂y ∂x / or fxy .
( iv ) The partial derivative of ∂f / ∂x w . r . t . ‘ x ’ is denoted by ∂ 2 f / ∂y ∂x / or fyx .
Note ∂ 2 f / ∂x ∂y = ∂ 2 f / ∂y ∂x
These four are second order partial derivatives . Euler ’ s Theorem on Homogeneous Function If f ( x , y ) be a homogeneous function in x , y of degree n , then x (& partf / ∂x ) + y (& partf / ∂y ) = nf