Prerequisite concepts
Difference / Change
Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.
Conceptually, a derivative can be thought of as how much one value changes as another value changes. FIXME: Representations
One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.
The derivative of a constant is always zero. This can be understood by thinking of 'The derivative is a ratio of small changes,' as the amount by which a constant changes is zero.
One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.