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Constant Multiples of Functions

What happens to the derivative of a function if we multiply the function by some constant?

Substitute image

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

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?

In general

Hopefully you have noticed that multiplying a function by a constant just multiplies the derivative by the same constant, or (d/dx)k*f(x)=k*f'(x).

Other differentiation topics

Acknowledgements

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