As we have seen on previous pages, the derivative is recording
how a function changes over a very small scale.
This page introduces one important fact that will be
used repeatedly in this course.
Tangent Line Approximation
If
is differentiable at
, then for
small values of
, the following is a good approximation:
Questions:
The tangent line approximation is fundamental for it
underlies every application of the derivative. Basically, it is
telling us how to approximate any function, which could be very
complicated, by a linear function, which is very easy to work with.
We'll see how to use it as we proceed. For now, you should try to
understand what it is saying and why it works.
Let's consider an
example so that we might understand it better.

An Example
The power of this approximation is not so apparent from this
examplenamely because the original function
is fairly simple. However, if
were a
more complicated function or a function which we did not yet know,
it would be very convenient to approximate
it by a linear function
. This
kind of reasoning will be important as we explore
derivatives further.
