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

This device cannot display Java animations. The above is a substitute static image

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

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 .

Related differentiation topics

• Constant, Line, and Power Functions
• Exponential Functions
• Trigonometric Functions
• Constant Multiple
• Combinations: Sum, and Difference
• Combinations of Functions: Product and Quotient
• Composition of Functions (the Chain Rule)
• Transformations of Functions
• Inverses of Functions
• Hyperbolic Functions
• Linear Approximation
• Mean Value Theorem

Acknowledgements

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