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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 f(a+h)-f(a)/h This ratio 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 quotient 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:

deriv. at a point.

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

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This device cannot display Java animations. The above is a substitute static image
See About the calculus applets for operating instructions.

1. A Parabola

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.

2. A sine curve

2. 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

3. A exponential function

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.

4. A hyperbola

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.

Explore

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

Other differentiation topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.