Final answer:
To find the first partial derivatives for the given function, use the quotient rule to differentiate with respect to x and y separately.
Step-by-step explanation:
To find the first partial derivatives for the function f (x,y) = √xy/ x-y, we need to differentiate the function with respect to x and y separately. Let's begin by finding the partial derivative with respect to x, denoted as ∂f/∂x.
To differentiate √xy/ x-y with respect to x, we use the quotient rule:
∂f/∂x = [(x-y)(1/2)(y) - (√(x)y)] / (x-y)^2
Simplifying this expression gives us the first partial derivative with respect to x. Similarly, to find the partial derivative with respect to y, denoted as ∂f/∂y, we differentiate the function with respect to y:
∂f/∂y = [(x-y)(1/2)(x) - (√(x)y)] / (x-y)^2
Simplifying this expression gives us the first partial derivative with respect to y.