Differentiation is about how small changes in one quantity influence other quantities. This “ratio of small changes viewpoint is often helpful in setting up problems involving differentiation, and will be especially useful later for partial derivatives.

For example, how do you determine how fast are you going? If you know how far you went during a given time interval, you can divide these quantities to determine your average speed during that interval. Repeat this computation over shorter and shorter time periods. Those are still, strictly speaking, average speeds, but they are better and better approximations to your instantaneous rate of change.

Equivalently, draw a graph showing your position as a function of time. If you connect any two points on this graph, the horizontal “distance” between them represents the amount of time it took to get from one to the other, and the vertical “distance” between them represents how far apart they are. Dividing these two quantities again yields your average speed for that part of your journey, which is also clearly the slope of the secant line connecting the two points. Repeat this computation over shorter and shorter time periods, and the slope of the secant line becomes a better and better approximation to the slope of the tangent line, which gives the instantaneous rate of change.

In either case, the construction outlined above yields your speed at a single point, but you can repeat the construction at every point. Derivatives are functions!

The constructions above are usually expressed in terms of limits. If your position is given by $y=2x^2$, then your speed is constructed by calculating \begin{equation} \frac{dy}{dx} = \lim_{\Delta x\to0} \frac{2(x+\Delta x)^2 - 2x^2}{\Delta x} = 4x \end{equation} where some algebra is required to obtain the final answer. The notation $\frac{dy}{dx}$ emphasizes that derivatives are built from ratios of the small quantities $\Delta y$ and $\Delta x$. Note the dependence on $x$; your speed is not constant.

Limits are rarely used when computing derivatives in practice. Instead, one derives and then typically memorizes a few basic rules, such as the Power Rule, which says that \begin{equation} \frac{d}{dx}(x^n) = nx^{n-1} \end{equation} and the Product Rule, which says that \begin{equation} \frac{d}{dx}(fg) = f \frac{dg}{dx} + g \frac{df}{dx} \end{equation} where $f$ and $g$ are arbitrary functions of $x$, that is, $f=f(x)$ and $g=g(x)$.