What does an exterior (second) derivative such as in $d^2=0$ have to do with second derivatives as in single- or multi-variable calculus? Is this a correct start:
Calculus derivatives are good for Taylor expansions (and thus optimization), and curvature.
Exterior derivatives are needed for integration — and are essential for generalizing the Fundamental Theorem of Calculus (i.e., Stokes theorem).
Best Answer
Good question! Here's a start. The ordinary derivative in one-variable calculus is a Lie derivative along a special vector field on $\mathbb{R}$; in particular, it is not a special case of the exterior derivative. The exterior derivative is instead some kind of "universal derivative": it records all of the information you would need to determine the derivative of a function along any vector field, for example. In particular, unlike the ordinary derivative, the exterior derivative of a function is a different kind of object, namely a $1$-form. Roughly speaking, a $1$-form is "the kind of thing that pairs with a vector field to return a number," so you can see the relationship there to what I said above.