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

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.

The example shows the line

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.

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 .

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

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