Partial Derivatives Approximation

Question

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation}

\begin{equation} \frac{\partial f(x,y)}{\partial y} \approx \frac {f(x,y+\delta y)-f(x,y)}{\delta x} \end{equation}

Is the following approximation true? \begin{equation} \frac{\partial ^2 f(x,y)}{\partial x \partial y} \approx \frac {f(x+ \delta x,y+\delta y)-f(x,y+\delta y)-f(x+ \delta x,y)+f(x,y)}{\delta x \delta y} \end{equation}

Answer 1

A useful formula

$$\frac{\partial^{2}f}{\partial x \partial y} \approx \frac{f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)}{4hk}$$

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