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.
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.