I thought about a way to simply explain the 2nd order approximation of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ around a point $(x_0,y_0)$.
First we take the linear approximation of $f$:
$$f(x,y) \approx f(x_0,y_0) + \frac{\partial f}{\partial x} \cdot \Delta x + \frac{\partial f}{\partial y} \cdot \Delta y $$
where all the partial derivatives are evaluated at $(x_0,y_0)$ and $\Delta x = x- x_0$, $\Delta y = y - y_0$.
Then I thought that for the second order approximation, we approximate the partial derivative themself with a first order approximation:
$$\frac{\partial f}{\partial x}(x,y) \approx \frac{\partial f}{\partial x}(x_0,y_0) + \frac{\partial^2 f}{\partial^2 x} \cdot \Delta x + \frac{\partial f}{\partial y \partial x} \cdot \Delta y $$ $$\frac{\partial f}{\partial y}(x,y) \approx \frac{\partial f}{\partial y}(x_0,y_0) + \frac{\partial^2 f}{\partial x \partial y} \cdot \Delta x + \frac{\partial f}{ \partial^2 y} \cdot \Delta y $$
And then putting this in the approximation of $f(x,y)$ we would get:
$$f(x,y) \approx f(x_0,y_0) + \left[ \frac{\partial f}{\partial x}(x_0,y_0) + \frac{\partial^2 f}{\partial^2 x} \cdot \Delta x + \frac{\partial f}{\partial y \partial x} \cdot \Delta y \right] \cdot \Delta x + \left[ \frac{\partial f}{\partial y}(x_0,y_0) + \frac{\partial^2 f}{\partial x \partial y} \cdot \Delta x + \frac{\partial f}{ \partial^2 y} \cdot \Delta y \right] \cdot \Delta y \\ = f(x_0,y_0) + \frac{\partial f}{\partial x} \cdot \Delta x + \frac{\partial f}{\partial y} \cdot \Delta y + \frac{\partial^2 f}{\partial^2 x } \cdot \Delta x^2 + 2 \frac{\partial^2 f}{\partial x \partial y} \cdot \Delta x \Delta y + \frac{\partial^2 f}{\partial^2 y} \cdot \Delta y^2 $$
But the real formula is $$ f(x,y) \approx f(x_0,y_0) + \frac{\partial f}{\partial x} \cdot \Delta x + \frac{\partial f}{\partial y} \cdot \Delta y + \frac{1}{2} \left( \frac{\partial^2 f}{\partial^2 x } \cdot \Delta x^2 + 2 \frac{\partial^2 f}{\partial x \partial y} \cdot \Delta x \Delta y + \frac{\partial^2 f}{\partial^2 y} \cdot \Delta y^2 \right) $$
Where si my reasoning wrong? Can it be corrected to get to the correct result or is it completely wrong?