Newton's method for estimating roots of functions
is an early numerical application of
the first derivative, often visualized with graphs.

[See also the bisection method in CCD.IVT The Intermediate Value Theorem-The Intermediate Value Theorem and The Bisection Method]

The differential or linear estimation is another early numerical application of the first derivative, often visualized with graphs. Again an alternative that presents a different connection to to the derivative using motion is visualized with mapping diagrams.