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 $x$.

Difference / Change The derivative relates how much $f$ changes as $x$ changes The derivative can be approximated by the slope of a secant line The derivative is a limit The derivative is a function The derivative of a constant is zero The derivative is a linear operator Power law The derivative at a cusp is undefined Variables can be held constant Product rule Single variable chain rule You can flip a derivative "With respect to what" matters The magnitude of the derivative at a point is the slope of a tangent line at that point

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.

Equations can be 'zapped with d' to relate differentials Individual differentials can be manipulated algebraically Total differentials are linear

Visualizing Divergence

The divergence is related to the total flux through a closed surface The divergence is a function The divergence is a local quantity

Visualizing Curl