We have shown that
\[\frac{d}{d x} \left(\right. x^{2} \left.\right) = 2 x \text{and} \frac{d}{d x} \left(\right. x^{1 / 2} \left.\right) = \frac{1}{2} x^{− 1 / 2} .\]
At this point, you might see a pattern beginning to develop for derivatives of the form \(\frac{d}{d x} \left(\right. x^{n} \left.\right) .\) We continue our examination of derivative formulas by differentiating power functions of the form \(f \left(\right. x \left.\right) = x^{n}\) where \(n\) is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, \(\frac{d}{d x} \left(\right. x^{3} \left.\right) .\) As we go through this derivation, note that the technique used in this case is essentially the same as the technique used to prove the general case.
Example
3.18
Differentiating \(x^{3}\)
Find \(\frac{d}{d x} \left(\right. x^{3} \left.\right) .\)
Solution
\[\frac{d}{d x} \left(\right. x^{3} \left.\right) & = \underset{h \rightarrow 0}{\text{lim}} \frac{\left(\left(\right. x + h \left.\right)\right)^{3} - x^{3}}{h} & & & \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - x^{3}}{h} & & & \begin{matrix}\text{Notice that the first term in the expansion of} \\ \left(\left(\right. x + h \left.\right)\right)^{3} \text{is} x^{3} \text{and the second term is} 3 x^{2} h . \text{All} \\ \text{other terms contain powers of} h \text{that are two or} \\ \text{greater}.\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{3 x^{2} h + 3 x h^{2} + h^{3}}{h} & & & \begin{matrix}\text{In this step the} x^{3} \text{terms have been cancelled}, \\ \text{leaving only terms containing} h .\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{h \left(\right. 3 x^{2} + 3 x h + h^{2} \left.\right)}{h} & & & \text{Factor out the common factor of} h . \\ & = \underset{h \rightarrow 0}{\text{lim}} \left(\right. 3 x^{2} + 3 x h + h^{2} \left.\right) & & & \begin{matrix}\text{After cancelling the common factor of} h , \text{the} \\ \text{only term not containing} h \text{is} 3 x^{2} .\end{matrix} \\ & = 3 x^{2} & & & \text{Let} h \text{go to 0}.\]
Checkpoint
3.12
Find \(\frac{d}{d x} \left(\right. x^{4} \left.\right) .\)
As we shall see, the procedure for finding the derivative of the general form \(f \left(\right. x \left.\right) = x^{n}\) is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate \(f \left(\right. x \left.\right) = x^{3} ,\) the power on \(x\) becomes the coefficient of \(x^{2}\) in the derivative and the power on \(x\) in the derivative decreases by 1. The following theorem states that the power rule holds for all positive integer powers of \(x .\) We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of \(x\) and then to arbitrary powers of \(x .\) Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as \(f \left(\right. x \left.\right) = 3^{x} .\)
Theorem
3.3
The Power Rule
Let \(n\) be a positive integer. If \(f \left(\right. x \left.\right) = x^{n} ,\) then
\[f^{'} \left(\right. x \left.\right) = n x^{n - 1} .\]
Alternatively, we may express this rule as
\[\frac{d}{d x} x^{n} = n x^{n - 1} .\]
Proof
For \(f \left(\right. x \left.\right) = x^{n}\) where \(n\) is a positive integer, we have
\[f^{'} \left(\right. x \left.\right) = \underset{h \rightarrow 0}{\text{lim}} \frac{\left(\left(\right. x + h \left.\right)\right)^{n} - x^{n}}{h} .\]
\[\begin{aligned} \text{Since} \left(\left(\right. x + h \left.\right)\right)^{n} = x^{n} + n x^{n - 1} h + \left(\right. n \\ 2 \left.\right) x^{n - 2} h^{2} + \left(\right. n \\ 3 \left.\right) x^{n - 3} h^{3} + \ldots + n x h^{n - 1} + h^{n} , \end{aligned}\]
we see that
\[\begin{aligned} \left(\left(\right. x + h \left.\right)\right)^{n} - x^{n} = n x^{n - 1} h + \left(\right. n \\ 2 \left.\right) x^{n - 2} h^{2} + \left(\right. n \\ 3 \left.\right) x^{n - 3} h^{3} + \ldots + n x h^{n - 1} + h^{n} . \end{aligned}\]
Next, divide both sides by h:
\[\begin{aligned} \frac{\left(\right. x + h \left.\right)^{n} - x^{n}}{h} = \frac{n x^{n - 1} h + \left(\right. n \\ 2 \left.\right) x^{n - 2} h^{2} + \left(\right. n \\ 3 \left.\right) x^{n - 3} h^{3} + \ldots + n x h^{n - 1} + h^{n}}{h} . \end{aligned}\]
Thus,
\[\begin{aligned} \frac{\left(\right. x + h \left.\right)^{n} - x^{n}}{h} = n x^{n - 1} + \left(\right. n \\ 2 \left.\right) x^{n - 2} h + \left(\right. n \\ 3 \left.\right) x^{n - 3} h^{2} + \ldots + n x h^{n - 2} + h^{n - 1} . \end{aligned}\]
Finally,
\[f^{'} \left(\right. x \left.\right) & = \underset{h \rightarrow 0}{\text{lim}} \left(\right. n x^{n - 1} + \left(\right. \begin{matrix}n \\ 2\end{matrix} \left.\right) x^{n - 2} h + \left(\right. \begin{matrix}n \\ 3\end{matrix} \left.\right) x^{n - 3} h^{2} + \ldots + n x h^{n - 2} + h^{n - 1} \left.\right) \\ & = n x^{n - 1} .\]
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Example
3.19
Applying the Power Rule
Find the derivative of the function \(f \left(\right. x \left.\right) = x^{10}\) by applying the power rule.
Solution
Using the power rule with \(n = 10 ,\) we obtain
\[f ' \left(\right. x \left.\right) = 10 x^{10 - 1} = 10 x^{9} .\]
Checkpoint
3.13
Find the derivative of \(f \left(\right. x \left.\right) = x^{7} .\)