Partial derivative

Definition at a point

Generic definition

Suppose $f$ is a function of more than one variable, where $x$ is one of the input variables to $f$. Fix a choice $x=x_0$ and fix the values of all the other variables. The partial derivative of $f$ with respect to $x$, denoted $\partial f/\partial x$, or $f_x$, is defined as the derivative at $x_0$ of the function that sends $x$ to $f$ at $x$ for the same fixed choice of the other input variables.

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x,y$, i.e., the domain of $f$ is a subset of ${\mathbb{R}}^2$. We define the partial derivatives as follows:

• Partial derivative with respect to $x$:

$\frac{\partial f(x,y)}{\partial x}{|}_{(x,y)=(x_0,y_0)}=\frac{d}{dx}f(x,y_0){|}_{x=x_0}$

In words, it is the derivative at $x=x_0$ of the function $x\mapsto f(x,y_0)$.

This partial derivative is also denoted $f_x(x_0,y_0)$ or $f_1(x_0,y_0)$.

• Partial derivative with respect to $y$:

$\frac{\partial f(x,y)}{\partial y}{|}_{(x,y)=(x_0,y_0)}=\frac{d}{dx}f(x_0,y){|}_{y=y_0}$

In words, it is the derivative at $y=y_0$ of the function $y\mapsto f(x,y_0)$.

This partial derivative is also denoted $f_y(x_0,y_0)$ or $f_2(x_0,y_0)$.

For a function of multiple variables

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