## Mathematical Analysis## Differential Calculus |

## What are the two ways of looking at the derivative? |

There are two major ways of looking at the derivative—the geometrical (or the slope of a curve) and the physical (the rate of change). The derivative was historically developed from finding the tangent line to a curve at a point (geometrically); it eventually became the study of the limit of a quotient usually seen as the change in *x* and *y* (Δ*y*/Δ*x*). Even today, mathematicians debate which is the best way to describe a derivative.

Geometrically, after determining the slope of a straight line through two points on a graph of a function, and the limit where the change in *x* approaches zero, the ratio becomes the derivative *dy/dx*. This represents the slope of a line that touches the curve at a single point—or the tangent line.

Physically, the derivative of *y* with respect to *x* describes the rate of change in *y* for a change in *x*. The independent variable, in this case *x*, is often expressed as time. For example, velocity is often expressed in terms of *s*, the distance traveled, and *t*, the elapsed time. In terms of average velocity, it can be expressed as Δ*s*/Δ*t*. But for instantaneous velocity, or as Δ*t* gets smaller and smaller, we need to use limits—or the instantaneous velocity at a point *B* is equal to: