Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 13/Partial Derivatives

Partial Derivatives

Let $\map f {x, y}$ be a function of the two variables $x$ and $y$. Then we define the partial derivative of $\map f {x, y}$ with respect to $x$, keeping $y$ constant, to be:

$13.58-59$: Definition of Partial Derivative

$13.58$: $\dfrac {\partial f} {\partial x} = \ds \lim_{\Delta x \mathop \to 0} \dfrac {\map f {x + \Delta x, y} - \map f {x, y} } {\Delta x}$

Similarly the partial derivative of $\map f {x, y}$ with respect to $y$, keeping $x$ constant, is defined to be:

$13.59$: $\dfrac {\partial f} {\partial y} = \ds \lim_{\Delta y \mathop \to 0} \dfrac {\map f {x, y + \Delta y} - \map f {x, y} } {\Delta y}$

Partial derivatives of higher order can be defined as follows.

$13.60-61$: Definition of Second Partial Derivative

$13.60.1$: $\dfrac {\partial^2 f} {\partial x^2} = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }$

$13.60.2$: $\dfrac {\partial^2 f} {\partial y^2} = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }$

$13.61.1$: $\dfrac {\partial^2 f} {\partial x \partial y} = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }$

$13.61.2$: $\dfrac {\partial^2 f} {\partial y \partial x} = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }$

By Clairaut's Theorem, $13.61.1$ and $13.61.2$ will be equal if the function and its partial derivatives are continuous, that is, in such case the order of differentiation makes no difference.

The differential of $\map f {x, y}$ is defined as:

$13.62$: Definition of Differential for Multi-Variable Functions: $\d f = \dfrac {\partial f} {\partial x} \rd x + \dfrac {\partial f} {\partial y} \rd y$

where $\d x = \Delta x$ and $\d y = \Delta y$.

Extension to functions of more than two variables are exactly analogous.