Suppose $F(x,y)$ is a function of two variables satisfying $F(0,0)=0$. By differentiating some expressions, I obtained the identity
$$ \frac{ \partial F}{\partial x}(x_0, y_0) = \int_0^1 \frac{\partial^2 F}{\partial x^2} (tx_0, ty_0) tx_0 + \frac{\partial F}{\partial x} (tx_0, ty_0) + \frac{\partial^2 F}{\partial y \partial x} (tx_0, ty_0) t y_0 ~~ {\bf dt}$$
Specifically, I got this by differentiating the identity $F(x,y) = \int_0^1 \frac{\partial F}{\partial t} (tx,ty) dt$. Is there any other way to see that the above identity holds, perhaps something more direct than the derivation I did?