Differentiation- Urgent help needed

Question

let 𝑈(𝑥, 𝑦) be defined for all x > 0, y > 0 by
𝑈(𝑥, 𝑦) = ln(𝑥^a+y^a)-Ln(y^a) where 𝛼 is a positive constant

find U'1(x,y), U'2(x,y) and U''12(x,y)

Answer 1

This question appears to be about partial derivatives of the function U(x,y)

 

I will assume that U'1(x,y) means the partial derivative of U with respect to x 

and that U'2(x,y) means that the partial derivative of U with respect to y and that

U''12  means the mixed second partial derivative of U.   

 

The usual notation for these would be:  U_x  U_y  and U_xy

 

The function U can be rewritten as:  U = ln[ (x/y)^a +1 ]

 

With this it is straightforward to get:

 

 U_x  = (a/x) (x/y)^a /  [ (x/y)^a +1 ]     and

 

 U_y = - (a/y) (x/y)^a  / [ (x/y)^a +1]

 

 The mixed second partial can be obtained by differentiating U_y  with respect to x.  The algebra here is a bit messy but there is a nice cancellation giving   

 

   U_xy  = - a² (x/y)^a  (1/x) (1/y)  / [ (x/y)^a + 1 ]²