How can I use this level curve diagram to estimate these partial derivatives? How can I use this level curve diagram:

To estimate these partial derivatives?
$f_{x}(1,1)$,
$f_{y}(1,1)$,
$f_{xx}(1,1)$,
$f_{yy}(1,1)$ and
$f_{xy}(1,1)$
Thanks in advance
 A: (Edited)
(i) The following is a very pedestrian approach:
When you want an estimate for $f_x(x,y)$ at some point $P=(x,y)$ in your diagram draw a horizontal through $P$. It will intersect two successive level lines at $P_l=(x-h_l,y)$ and $P_r=(x+h_r,y)$, where $h_l$, $\>h_r\geq 0$ and $\Delta x:=h_l+h_r>0$. You then have approximatively
$$f_x(x,y)\doteq{\Delta f\over\Delta x}\ ,$$
where $\Delta f$ denotes the value difference of $f$ between successive level lines. – Similarly for $f_y(x,y)$.
For $f_{xx}$ I shall assume that $P=(x,y)$ lies on a level line. There will be two neighbouring level lines intersecting the horizontal through $P$ at $P_l$ and $P_r$ as above. This provides you with the estimates
$$f_x\bigl(x-{h_l\over2},y\bigr)\doteq{\Delta f\over h_l}, \quad f_x\bigl(x+{h_r\over2},y\bigr)\doteq{\Delta f\over h_r}\ ,$$
so that you finally obtain
$$f_{xx}(x,y)\doteq {2\over h_lh_r}\ {h_l-h_r\over h_l+h_r}\ \Delta f\ .$$
(ii) The following is more sophisticated:
Assume you want to find the partial derivatives of $f$ at some point $(p,q)$ in your diagram. This means you want to know the Taylor coefficients of $f$ at $(p,q)$: The function $f$ has  an expansion of the form
$$f(p+X,q+Y)=a+ (b_1 X+b_2 Y)+(c_1 X^2+2c_2 XY+c_3Y^2)+(d_1 X^3+\ldots)+\ldots\ .$$
Here the $a=f(p,q)$, $b_1=f_x(p,q)$, $\ldots$, $\>d_1={1\over6}f_{xxx}(p,q)$.
To determine values for these coefficients proceed as follows: Choose $n$ (e.g. $n=9$) points $$(x_k,y_k)=(p+X_k,q+Y_k)$$ in the neighborhood of $(p,q)$ which lie on at least three different red curves. You then know the exact function values $z_k:=f(x_k,y_k)$ at these points. Use these data to determine the coefficients $a$, $\>b_i$, $\ldots$, in the above development. You will need third order terms when there are inflection points of the red curves involved. The resulting system of equations is an overdetermined linear system, as is usual in regression analysis. The methods developed there will give you an "optimal guess" for the coefficients $a$,$\>b_i$, $\ldots$,  whence for the partial derivatives of $f$ at $(p,q)$, in the sense of least squares.
