Notation

It is a bit cumbersome to always carry around the idea of a section when computing derivatives of multivariable functions. We now introduce the notation of partial derivatives. We call these partial derivatives because we are only differentiating the function in one direction. Thus, we only get a partial picture of how this function changes as the variables change. The following graph shows some arbitrary function of two variables and illustrates the slope in the x and y directions.

To denote a partial derivative, we use the symbol $\partial$ rather than d. Thus, $\frac{\partial f}{\partial x}$ is the partial derivative (read slope) in of f(x,y) in the x direction (ie. along a section y = c) and $\frac{\partial f}{\partial y}$ is the partial derivative of f in the y direction. It is also common to denote the derivatives as f_x and f_y, respectively.