Consider the case of finding the average rate of change between two points on the Cartesian plane:
$$ \begin{equation}\begin{aligned} m&=\frac{\Delta y}{\Delta x}\\ &=\frac{y_2-y_1}{x_2-x_1}\\ \end{aligned}\end{equation} $$If we bring the two points infinitesimally close to each other, we get the notation: $$ \begin{equation}\begin{aligned} m&=\frac{dy}{dx}\\ \end{aligned}\end{equation} $$
This is the instantaneous rate of change a.k.a. the derivative of $y$ with respect to $x$. This is the gradient function of the graph of $y$ plotted against $x$.