Langara College - Department of Mathematics and Statistics

## 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 on-line animations and applets which illustrate how this new function is related to the given one

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 web-based illustrations of these and related concepts,