Concept: The derivative is a limit

Prerequisite concepts

The derivative is a ratio of small changes

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 relates how much

$f$ changes as

$x$ changes

lower anchor MTH 251

The derivative is a limit

Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Zapping with d

Differentials are small changes or differences

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.

Visualizing Divergence

Visualizing Curl