# Derivative of a Power Function

## Power Functions

A basic (non-transformed) power function has the form of a variable base x raised to a constant power n. In this activity we let n be a rational number. Therefore, n = p/q where p is an integer and q is a natural number. You should start by graphing several examples of power functions. Make sure that you can answer the following series of questions about the graphs of basic power functions before going on to examine their derivatives.

What quadrant is always present in the graph and what quadrant is always missing? Why?

## Common Point(s)

What point is on every graph of a basic power function, regardless of the power?

What does the size of the power, n, have to do with the shape of the graph of a basic power function in Quadrant I?

## Symmetry

What do the even or odd nature of p and q tell us about the graph of a basic power function? Consider what happens with a negative input to help answer this question.

## x = 0?

What happens for an input of 0?

## Derivatives

Now that you have a good feel for the properties of the graphs of basic power functions, manipulate the values of p and q to get the various cases of possible graph shapes for the original power function. For each of these try to figure out the appropriate shape of the derivative function, based upon your knowledge of the relationships between the graphs of a function and its derivative. When you think you have identified the basic shape of the derivative function, check your work by clicking on the checkbox to show the Derivative.

## Power Function Rule for Derivatives

The Power Function Rule for Derivatives is given above when you check the Derivative checkbox. To find the derivative of a power function, we simply bring down the original power as a coefficient and we subtract 1 from the power to get the new power. Therefore, the derivative of a power function is a constant times a basic power function. The graph of the derivative of a power function will also be the graph of a basic power function that has been modified by multiplying by a constant. Recall that multiplying a function by a constant results in a vertical stain (expansion |n|>1 or compression |n|<1 from the x-axis) and also a reflection over the x-axis if n is negative.

## Proof

One can use the definition of the derivative to prove this differentiation shortcut rule for integer powers of n. However, to prove the result in general requires that we first know the constant function rule, chain rule, exponential rule, logarithm rule, constant multiple rule, and sum rule, which we can apply to form a more general power/exponential rule. It turns out that the power function derivative rule works for any constant power. Note that this is for a variable base and a constant power. Do not confuse this with an exponential function, which has a constant base and variable exponent, or with a function with a variable base and a variable exponent. Be sure that you use derivative rules only where they apply.

## Warning

When I tried to use the derivative command in GeoGebra on these functions, it did not accurately graph the derivatives when x was negative in some cases.