The Definition of Differentiation
The Definition of Differentiation
We want to find the slope of the tangent line to a graph at the point
P. We can approximate the slope by drawing a line through the
point P and another point nearby, and then finding the slope
of that line, called a secant line. The slope of a line is determined
using the following formula (m represents slope) :
Let P = (x,y) and Q = (a,b). Let
and
. Then the slope of
the line
is
Now, we chose an arbitrary interval to be
. How
does the size of
affect our estimate of the slope
of the tangent line? The smaller
is, the more
accurate this approximation is. There is a wonderful animation of this by
Douglas
Arnold. Look at it
here.
You can see on the left of the animation how
decreases, causing the secant line the approach the tangent, where it
zooms in on the right. Another animation of this (also from Douglas
Arnold) is
here.
What we want to do is decrease the size of
as
much as possible. We do this by taking the limit as
approaches zero. In the limit, assuming the limit
exists, we will find the exact slope of the tangent line to the curve at
the given point. This value is the derivative.
There are a few different, but equivalent, versions of this definition.
In general, h is sometimes used in place of
. Or
is used in place of
. This leads to three commonly used ways of expressing the
definition of the derivative: