# Definition:Partial Derivative/Second Derivative

## Definition

Let $f \left({x, y}\right)$ be a function of the two independent variables $x$ and $y$.

The second partial derivatives of $f$ with respect to $x$ and $y$ are defined and denoted by:

$(1): \quad \dfrac {\partial^2 f}{\partial x^2} = \dfrac {\partial}{\partial x} \left({\dfrac {\partial f}{\partial x}}\right)$
$(2): \quad \dfrac {\partial^2 f}{\partial y^2} = \dfrac {\partial}{\partial y} \left({\dfrac {\partial f}{\partial y}}\right)$
$(3): \quad \dfrac {\partial^2 f}{\partial x \partial y} = \dfrac {\partial}{\partial x} \left({\dfrac {\partial f}{\partial y}}\right)$
$(4): \quad \dfrac {\partial^2 f}{\partial y \partial x} = \dfrac {\partial}{\partial y} \left({\dfrac {\partial f}{\partial x}}\right)$