Langara College  Department of Mathematics and Statistics
Internet Resources for the Calculus Student
Interpretation of Derivatives
The derivative of a function
gives us the rate of change of
the given function, and this corresponds geometrically to the slope of
the tangent line to the graph of the function. This may vary from point
to point on the graph, and so the slope value is just another function
of the position (this new function is what is called the "derivative" of
the function you started with). The following links are to online
animations and applets which illustrate how this new function is related
to the given one

This shows
the tangent line to a curve, varying as the contact point moves along the
curve.

Surfing(Derivatives)@ies

Guessing
the Derivative from a graph

Pickthe
correct derivative! by David Yuen
The value of the derivative (eg either from a graph or a formula) tells
us how fast the function is increasing or decreasing. So if we are given
a starting point we can determine the function itself if we are given its
derivative (just as you might determine the position of a ship at sea by
keeping track of how fast it had been moving)  see Antiderivatives
.
And at any extreme (high or low) point of a function, if the graph has
a well defined slope then that slope must be zero. So derivatives
are useful for locating maxima and minima of functions. ie for Optimization
problems. See also derivCubics@ies

Rate of change of slope (ie derivative of derivative or "second derivative")
corresponds to curvature , so knowledge of the derivative and second derivative
of a function can help us to produce a quick qualitatively correct picture
of its graph.(See also CurvatureCircle@ies,and
LnGraphCircle@ies)


What
does the derivative tell us about a function?(@ubc)

If you have come across any good webbased illustrations of these and
related concepts,
please do let
us know and we will add them here.