Partial derivatives | Wiskunde op Tilburg University

Introduction: When we consider a function of one variable

$y(x)$ , then that function has a derivative

$y'(x)$ . Functions of two variables

$z(x,y)$ also have derivatives, but we have to specify with respect to which variable we differentiate the function. The other variable is held constant.

Definition: The derivative of the function

$z(x,y)$ with respect to

$x$ at fixed

$y$ is called the partial derivative of

$z(x,y)$ with respect to

$x$ and denoted by

$z'_x(x,y).$
The derivative of the function

$z(x,y)$ with respect to

$y$ at fixed

$x$ is called the partial derivative of

$z(x,y)$ with respect to

$y$ and denoted by

$z'_y(x,y).$

Remark 1: Other notations for the partial derivatives of

$z(x,y)$ with respect to

$x$ and

$y$ are

$\dfrac{\partial}{\partial x} z(x,y), \quad \dfrac{\partial z}{\partial x}(x,y), \quad \dfrac{\partial}{\partial y} z(x,y) \quad \text{and} \quad \dfrac{\partial z}{\partial y}(x,y).$

Remark 2: Sometimes these partial derivatives are also abbreviated to

$z'_x$ and

$z'_y$ .

Remark 3: The Derivatives of elementary functions, and the rules of differentiation and the chain rule for functions of one variable, can also be applied when differentiating a function of two (or more) variables.