derivative of a function in two variables notation

Question

Quick question. What does $$\frac{\delta^2 f}{\delta x \delta y}$$ mean? Is it to multiply $\frac{\delta f}{\delta x}$ with $\frac{\delta f}{\delta y}$ or something else and if so, what?

Answer 1

It is something else

$\frac{\delta}{\delta x }$ is an operation applied to a function, namely taking the partial derivative with respect to $x$. So $\frac{\delta}{\delta x }(f)$ or $\frac{\delta f}{\delta x }$ is the partial derivative of $f$ with respect to $x$

$\frac{\delta}{\delta y }$ is another operation, namely taking the partial derivative with respect to $y$

$\frac{\delta^2 f}{\delta x \delta y}$ is shorthand for $\frac{\delta}{\delta x }\left(\frac{\delta}{\delta y }(f)\right)$, namely taking the partial derivative with respect to $x$ of the partial derivative of $f$ with respect to $y$

For example if $f(x,y)=(x^2+y^2)^2$ then $\frac{\delta f}{\delta y } = 4y(x^2+y^2)$ and so $\frac{\delta^2 f}{\delta x \delta y} = 8xy$. By contrast $\frac{\delta f}{\delta x } \times \frac{\delta f}{\delta y } = 16xy(x^2+y^2)^2$