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
Find all the second partial derivatives.
f(x, y) = (x^6)y − (2x^3)y^2
3 answers
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
∂^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
∂^2/∂x∂y [f(x, y)] = 6x^5 - 12x^2 y