Derivative at a Point
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
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.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.