Find all partial derivatives?
v = (xy)/(x-y)
vxx=
vxy=
vyx=
vyy=
To find the partial derivatives of the function v = (xy)/(x-y), we need to differentiate the function with respect to each variable separately and assume that any other variable is a constant.
First, we'll find the partial derivative vxx (the second partial derivative with respect to x). To differentiate v with respect to x, treat y as a constant:
v = (xy)/(x-y)
To differentiate (xy) with respect to x, we can use the product rule:
(d/dx)(xy) = x(dy/dx) + y(dx/dx)
= x(dy/dx) + y
Next, we need to find the partial derivative of (x-y) with respect to x:
(d/dx)(x-y) = 1 - (dy/dx)
Using the quotient rule, the partial derivative of v with respect to x (v_x) is:
v_x = [(x-y)(dy/dx) + y(1 - (dy/dx))] / (x-y)^2
Simplifying this expression:
v_x = [x(dy/dx) - y(dy/dx) + y - (dy/dx)y] / (x-y)^2
= (x - 2y(dy/dx)) / (x-y)^2
Thus, vxx = (x - 2y(dy/dx)) / (x-y)^2
To find vxy (the mixed partial derivative with respect to x and y), we need to differentiate the expression for v_x with respect to y:
vxy = d(v_x)/dy = d/dy[(x - 2y(dy/dx)) / (x-y)^2]
Using the quotient rule and the chain rule, we differentiate the numerator and denominator separately:
d/dy(x - 2y(dy/dx)) = -2(dy/dx)
d/dy(x-y)^2 = 2(x-y)(-1) = -2(x-y)
Applying the quotient rule, the expression for vxy becomes:
vxy = [(-2(dy/dx))(x-y) - (x - 2y(dy/dx))(-2(x-y))] / (x-y)^4
Simplifying this expression:
vxy = (-2(x-y)(dy/dx) + 2(x-y)(x - 2y(dy/dx))) / (x-y)^4
= (2(x-y)(-1 + (x - 2y(dy/dx)))) / (x-y)^4
= 2(x-y)(-1 + x - 2y(dy/dx)) / (x-y)^4
= 2(x-y)(x - 1 - 2y(dy/dx)) / (x-y)^4
Thus, vxy = 2(x-y)(x - 1 - 2y(dy/dx)) / (x-y)^4
Following the same steps, we can find vyy and vyx:
vyy = [(2(x-y)(dy/dx) + 2(x-y)(x - 2y(dy/dx)))(2(x-y)) - 2(x - 2y(dy/dx))^2] / (x-y)^6
= [2(x-y)(dy/dx) + 2(x-y)(x - 2y(dy/dx)))(2(x-y)) - 4(x - 2y(dy/dx))^2] / (x-y)^6
= 2(x-y)(dy/dx) + 2(x-y)(x - 2y(dy/dx)) - 2(x - 2y(dy/dx))^2 / (x-y)^5
vyx = [(x - 2y(dy/dx)) - 2(x-y)(-1 + x - 2y(dy/dx)) / (x-y)^3
= (x - 2y(dy/dx)) - 2(x-y)(-1 + x - 2y(dy/dx)) / (x-y)^3
Please note that these partial derivatives depend on the value of dy/dx, which we have assumed to be given. If you have a specific value for dy/dx, you can substitute it into the expressions above to find the numerical values of vxx, vxy, vyx, and vyy.