In the above applet, there is a pull-down menu at the top to select which function you would like to explore.
The selected function is plotted in the left window and its derivative on the right.
1. A line
The example shows the line
f (x) = kx
We know that the slope for the line f (x) = x is just 1. What does the k do to the slope? Move the k slider and you notice that the new slope is just k.
2. A parabola
Select the second example, showing a parabola multipled by k. When k > 1, what happens to the shape of the parabola? Move the k slider to see. What happens to the derivative?
Set x = 1 and k = 1 and notice the value of f '(1). Now, change k to 2; what happens to the derivative at x = 1? Change k to other values, and see if you can detect a pattern in what happens to the derivative.
3. A sine curve
Select the third example, showing a sine curve multiplied by a constant. What does changing k do to the derivative?
Hopefully you have noticed that multiplying a function by a constant just multiplies the derivative by the same constant, or