etc., if they exist. The process of finding such partial derivatives are called successive partial derivatives. If we differentiate u = f ( x,y ) partially with respect to x and again differentiating partially with respect to y , we obtain y u l i.e., ∂ ∂∂ u y x Similarly, if we differentiate u = f ( x,y ) partially with respect to y and again differentiating partially with respect to x , we obtain ∂ ∂ ∂ ∂ u y i.e., ∂ ∂∂ u x y NOTE If u ( x,y ) is a continuous function of x and y, then ∂ ∂∂ = ∂ ∂∂ u y x u x y . Homogeneous functions A function f ( x,y ) of two independent variables x and y is said to be homogeneous in x and y of degree n if f tx ty t f x y ( , ( , ) for t > .
. . Euler’s theorem and its applications Euler’s theorem for two variables: If u f x y ( , ) is a homogeneous function of degree n , then x u y u nu ∂ ∂ + ∂ ∂ Example . If u = x ( y–x ) + y ( x–y ), then show that ∂ ∂ + ∂ ∂ u u y = – ( x–y ) .
- - Applications of Differentiation u = x y – x + xy – y u = xy – x + y u = x + xy – y u u = – x – y + xy = − xy = – ( x – y ) Example . If u = log( x + y ), then show that u u = u = log( x + y ) u = x ( x ) = x ¶ ¶ u x = $ $ _ _