2.3 Partial derivatives

We consider $f(x,y)$ as a function of $x$ and $y$ separately, and investigate the rates of change of $f$ in the $x$ and $y$ directions. First we fix $y=b$, a constant, and consider $f(x,b)$, which now depends upon $x$ only.

We define the partial derivative of $f$ with respect to $x$ at $(a,b)$ to be the limit

when this limit exists. We use curly letters for partial derivatives to distinguish them from ordinary derivatives.

Now we fix $x=a$ and consider the rate of change of $f$ in the $y$ direction. The partial derivative of $f$ with respect to $y$ at $(a,b)$ is

whenever this limit exists.