The Handy Math Answer Book

What are the two waysof 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 devel-
oped 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 xand y
(y/x). Even today, mathematicians still debate which is the most useful and 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 xapproaches 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 ywith respect to xdescribes 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 s/t. But for instan-
taneous velocity, or as tgets smaller and smaller, we need to use limits—or the
instantaneous velocity at a point Bis equal to:

What are some examplesof derivatives of “simple” functions?

The following lists some derivatives of “simple” functions:

What are some simple derivativesas functions of the variablex?

There are simple derivatives as functions of the variable x. In this case, uand vare
functions of the variable x,and nis a constant:

dxdln x =^1 x

dx

dxnxnn= - 1

t

lim

t

s

3 " (^03)
3
dx
dy 1
dy
= dx

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MATHEMATICAL ANALYSIS

t

0 s

3

3