How to find $f_{xy}$ if only $f_y$ and $f_x$ are given?

Question

I am given the partial derivative of a function $f(x,y)$ with respect to $x$ is $4x^2-36$ and the partial derivative with respect to $y$ is $5y+20$. How to find $f_{xy}$ ?

Answer 1

$f_x = 4x^2 - 36 \to f = \int f_xdx = \int (4x^2-36)dx = \dfrac{4x^3}{3} - 36x + C(y)\to f_y = C'(y) = 5y+20 \to C(y) = \int (5y+20)dy = \dfrac{5y^2}{2} + 20y + D$. Thus:

$f(x,y) = \dfrac{4x^3}{3} - 36x+\dfrac{5y^2}{2} + 20y + D$. Can you continue?