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