We learn how to find the derivative of a power function.
The Power Rule
In this section we present the derivatives power functions. A power function is a function of the form .
We will investigate this rule in the case where n is a whole number. The key is to use the Binomial Theorem, which gives a formula for expanding powers of . Here are some examples of the Binomial Theorem:
The coefficients can be determined using a device called Pascal’s Triangle. Notice that the coefficient of the second term on the right hand side always matches the power on the left hand side. In general,
Also observe that as the powers of decrease, the powers of increase.
We will now use the definition of the derivative to discover the power rule for ourselves. Let , where n is a whole number. Then
We can rewrite the function as a power function, since and then we can use the power rule with . We get . The graph of the constant function is a horizonatal line which has slope 0. Hence the derviative should be 0. More generally, the derivative of any constant function should be 0.
The graph of a constant function is a horizontal line with slope 0. The graphs of , and are shown below. (To see , zoom out by clicking on the minus in the graphing window or by placing the cursor in the graphing window and using the scroll wheel on your mouse).
We can use the power rule with to obtain . The graph of the function is a straight line with slope , so the derivative should be . More generally, the graph of a function is a line with slope , so the derivative of should be .
The graph of is a line with slope . The graphs of , and are shown below. Which one is which? Click on the ¿¿ in the upper left hand corner of the graphing window to check your answer.
Note that this answer is just the slope of the line .
We now consider examples where the exponment is a fraction. Recall the defintion of rational exponents:
We can rewrite as and use the power rule with to obtain . This can then be rewritten in radical form : This result is used frequently, so it is best to remember it as
We can rewrite as and use the power rule with to obtain . This can be rewritten in radical form as
We first rewrite as (adding exponents) and then we use the power rule with to obtain . This can be rewritten in radical form as
Next, we look at some examples involving negative exponents. Recall the definition of negaive exponents:
We can rewrite as and use the power rule with to obtain We get . This can be rewritten without the negaive exponent as This result is used frequently, so it is best to remember it as
We can rewrite as and use the power rule with to obtain . This can be rewritten without the negative exponent as
We can rewrite as and use the power rule with to obtain . This can be rewritten without the negative exponent as
We can rewrite as (subtracting exponents). Now, we use the power rule with to obtain This can be rewritten without the negative exponent as