Help with Partial Derivative?

Question

Help please! What are the partial derivatives of the following:
a. -2xy/((x^2+y^2)^2) with respect to x
b. (1-((x^2-y^2)/(x^2+y^2)^2) with respect to y

Answer 1

a) d/dx[(-2x y)/((x^2+y^2)^2)]

=(-2y)d/dx[(x)/((x^2+y^2)^2)]

use product rule

=(-2y)[1/((x^2+y^2)^2) + x (-2)(2x)/(x^2+y^2)^3]

=(-2y)[1/((x^2+y^2)^2) - (4x^2)/(x^2+y^2)^3]

now simplify

=(-2y)(y^2-3x^2)/(x^2+y^2)^3


b) d/dy[1-(x^2-y^2)/(x^2+y^2)^2]

=d/dy[-(x^2-y^2)/(x^2+y^2)^2]

use product rule again

=(2y)/(x^2+y^2)^2 -(x^2-y^2)(-2)(2y)/(x^2+y^2)^3

=(2y)/(x^2+y^2)^2 +(4y)(x^2-y^2)/(x^2+y^2)^3

now simplify

=(-2)(y^3-3x^2y)/(x^2+y^2)^3

Answer 2

a) [-2y(3x^2-y^2)]/(x^2+y^2)^3

b) [-2y(y^2-3x^2)]/(y^2+x^2)^2