The following integral when solved by integration by parts comes out to be zero.
$$\int^{10}_{0}f(x,y)\dfrac{d^{2}[g(x,y)]}{dxdy}dx$$ $$=\left[ \int f(x,y)\dfrac{d^{2}[g(x,y)]}{dxdy}dx \right]^{10}_{0}$$ $$=\left[ f(x,y)\dfrac{d[g(x,y)]}{dy}-\int\dfrac{d[f(x,y)]}{dx}\dfrac{d[g(x,y)]}{dy}dx \right]^{10}_{0}$$ {integration by parts} $$=\left[ f(x,y)\dfrac{d[g(x,y)]}{dy}-\dfrac{d[g(x,y)]}{dy} \int\dfrac{d[f(x,y)]}{dx}dx \right]^{10}_{0}$$ {since $\dfrac{d[g(x,y)]}{dy}$ is a function of $y$ and can be taken outside integration w.r.t. $x$} $$=\left[ f(x,y)\dfrac{d[g(x,y)]}{dy}-\dfrac{d[g(x,y)]}{dy} f(x,y) \right]^{10}_{0}=0$$
This integral is coming out to be zero for any $f(x,y)$ or $g(x,y)$ at any $y$
CHECKING
Let $f(x,y)=xy$ and $g(x,y)=x^{2}y$ $$\dfrac{d^{2}[g(x,y)]}{dxdy}=\dfrac{d}{dx}\left[ \dfrac{d(x^{2}y)}{dy} \right]=\dfrac{d}{dx}(x^2)=2x$$
$$\int^{10}_{0}f(x,y)\dfrac{d^{2}[g(x,y)]}{dxdy}dx=\int^{10}_{0}(xy.2x) dx=2y\int^{10}_{0}x^{2} dx=2y\left[ \dfrac{x^{3}}{3} \right]^{10}_{0}$$ $$=2y\left[ \dfrac{1000}{3}-\dfrac{0}{3} \right]=\dfrac{2000}{3}y$$
Now this is not coming out to be zero at any $y$. Why is this contradiction happening? Am I somewhere wrong while doing integration by parts?
Well, $$ \frac{\partial g}{\partial y}(x,y) $$ is not a function only of $y$. It also dependes on $x$.