Partial Derivative

A partial derivative is deriving a function with respect to a variable, this derives only one variable while the other remains "constant"

Partial Derivative of f(a, b) in respect to x $$ f_{x}(a,b) = \lim_{ h \to 0 } \frac{f(a+h,b)-f(a,b)}{h} $$

Partial Derivative of f(a, b) in respect to y $$ f_{y}(a,b) = \lim_{ h \to 0 } \frac{f(a,b+h)-f(a,b)}{h} $$

Notations of Partial Derivative

Partial Derivatives are can be notated in a number of ways

The partial derivative can be taken multiple times

Differentiability

If all partial derivatives of a function are differentiable, then the function itself is differentiable

Partial differential equations are differential equations with partial derivatives

The heat equation, for example, gives temperature as a function of position and time

Mixed Partials are partial derivatives where derivatives in respect to more than one variable is taken

Mixed Partial of f in respect to x and y $$ f_{xy} = \frac{\partial^2f}{\partial x \partial y} $$

Order of Mixed Partials

The partial is equal to
Similarly, is equal to and

If the relevant mixed partials exists and are continuous around the particular position then