Partial derivative

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 $\mathbb {R} ^{2}$. 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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