Find all the second partial derivatives.
f(x, y) = (x^6)y − (2x^3)y^2
not bad, but
∂^2/∂x∂y f(x, y) = ∂/∂x ∂/∂y f(x, y) = ∂/∂y (6x^5 y - 6x^2 y^2)
= 6x^5 - 12x^2 y
You are correct, thank you for catching that mistake!
∂^2/∂x∂y [f(x, y)] = 6x^5 - 12x^2 y
To find the second partial derivatives of the function f(x, y) = (x^6)y - (2x^3)y^2, we need to find the partial derivatives with respect to x and y, and then take the partial derivatives of those results with respect to x and y again.
First, let's find the first partial derivative with respect to x:
∂/∂x [(x^6)y - (2x^3)y^2]
To take the partial derivative with respect to x, we treat y as a constant. The derivative of x^6 with respect to x is 6x^5, and the derivative of x^3 with respect to x is 3x^2. So we have:
∂/∂x [(x^6)y - (2x^3)y^2] = (6x^5)y - (6x^2)y^2
Next, let's find the first partial derivative with respect to y:
∂/∂y [(x^6)y - (2x^3)y^2]
To take the partial derivative with respect to y, we treat x as a constant. The derivative of y with respect to y is 1, so we have:
∂/∂y [(x^6)y - (2x^3)y^2] = x^6 - 2(2x^3)y
Now, let's find the second partial derivative with respect to x:
∂^2/∂x^2 [(6x^5)y - (6x^2)y^2]
To take the second partial derivative with respect to x, we treat y as a constant. The derivative of 6x^5 with respect to x is 30x^4, and the derivative of -6x^2 with respect to x is -12x. So we have:
∂^2/∂x^2 [(6x^5)y - (6x^2)y^2] = (30x^4)y - (-12x)y^2
= (30x^4)y + (12x)y^2
Finally, let's find the second partial derivative with respect to y:
∂^2/∂y^2 [x^6 - 2(2x^3)y]
To take the second partial derivative with respect to y, we treat x as a constant. The derivative of x^6 with respect to y is 0, and the derivative of -4x^3y with respect to y is -4x^3. So we have:
∂^2/∂y^2 [x^6 - 2(2x^3)y] = 0 - (-4x^3)
= 4x^3
Therefore, the second partial derivatives of f(x, y) are:
∂^2/∂x^2 = (30x^4)y + (12x)y^2
∂^2/∂y^2 = 4x^3
To find all the second partial derivatives, we need to take the partial derivative of f with respect to x and y twice, in different orders.
∂/∂x [f(x, y)] = 6x^5 y - 6x^2 y^2
∂^2/∂x^2 [f(x, y)] = 30x^4 y - 12xy^2
∂/∂y [f(x, y)] = x^6 - 4x^3 y
∂^2/∂y^2 [f(x, y)] = -8x^3
∂^2/∂x∂y [f(x, y)] = 6x^4 - 4x^3
Therefore, the second partial derivatives are:
∂^2/∂x^2 [f(x, y)] = 30x^4 y - 12xy^2
∂^2/∂y^2 [f(x, y)] = -8x^3
∂^2/∂x∂y [f(x, y)] = 6x^4 - 4x^3