Partial derivative

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Definition at a point

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 $\R^2$. 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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