This file is part of a distribution of the Calculus Applets website (http://www.calculusapplets.com) (v1.1) which has been reformatted for the needs of this OpenCourseware course.

We can generalize the concept of average velocity to any function, not
just to a position versus time function. We can define the *average rate
of change* of a function, *f*, on an interval from *a* to *a* + *h* as . This ration is called the *difference quotient* and
is the change in the output of the function divided by the change in the
input. We can similarly extend the concept of instantaneous velocity to the *instantaneous rate of change* of a function by taking the limit of
the difference qoutient as *h* approaches 0. Specifically, we define the
rate of change of function *f* at *a*, called the derivative of *f* at *a* and written *f '* (*a*), as:

.

The function *f* is said to be differentiable at *a* if this
limit exists.

Try the following:

- The applet initially shows a parabola. What is the derivative of this
function at
*x*= 1? The green line represents a secant connecting the points (1,1) and (1.9,3.61). The slope of this secant line is the average rate of change of the function over the interval from 1 to 1.9. As you drag the green dot towards the red dot, you are essentially decreasing*h*in the difference quotient, so the slope of the secant line approaches the derivative at 1. The red line is tangent to the curve at*x*= 1 and hence the slope of the red line is the derivative at 1.

- Click the zoom in button. What happens to the shape of the parabola
curve? Zoom in a few more times. How does the black parabola curve look,
relative to the red tangent line? As you zoom in on a point of a function
where there is a derivative, the curve of the function will look more and
more like a straight line, in particular like the tangent line.

- Select the second example from the drop down menu. This example shows a
sine curve with the horizontal axis in radians. What is the derivative of
this function at
*x*= π? Drag the green dot and notice how the slope of the secant line changes, approaching the slope of the tangent line. Hence the derivative of sin(*x*) at π is -1.

- Click the zoom in button a few times. Notice that the sine curve looks
more and more like the straight tangent line.

- Select the third example from the drop down menu. This shows an
exponential function. What is the derivative at
*x*= 0? Dragging the green dot, or looking at the slope of the red tangent line provides the answer. Zooming in makes the curve look like the tangent line.

- Select the fourth example, a hyperbola. What is the derivative at
*x*= 1? Dragging the green dot, or looking at the slope of the red tangent line provides the answer. Zooming in makes the curve look like the tangent line.

- You can also enter your own function into the "f(x)=" box.

For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.

© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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Citation: tdowney. (2009, August 06). Derivative at a Point. Retrieved October 22, 2014, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/elements-of-calculus-i/calculus-applets-website/Derivative%20at%20a%20Point.html.

Copyright 2012,
Thomas S. Downey.
This work is licensed under a
Creative Commons Attribution 3.0 License.