The Power Rule

We are often interested in making things simpler and more time-efficient in day-to-day life. Would you wash dishes by hand if you have a dishwasher? Or re-type a paragraph if you can copy and paste? We can say the same for mathematics!

Take the Derivative as an example. Finding the derivative of a function is one of the basic operations in Calculus. However, using limits can be time-consuming as there are many steps and lots of algebra are involved. Here's an example of that process.

Lots of steps, right? Rather than doing all these procedures, there are many formulas in calculus that we can use to find derivatives with fewer steps, saving us time and mental energy. These formulas are known as derivative rules, and one of these derivative rules is the power rule.

Power rule formula and examples

One of the primary functions found in calculus is the power function.

These functions are essential in Calculus for building more advanced functions, like polynomial functions or rational functions. We can find the derivative of a power function using what is known as the power rule. Let's take a look at it.

This rule can make finding derivatives in calculus much simpler! Let's take a look at some examples.

We can use the power rule in combination with other Differentiation Rules to find the derivative of a polynomial function. Let's look at an example of this process.

Deriving the power rule

To prove the power rule, we will look at the derivative of $f\left(x\right)=x^n$using limits. We need to find such a derivative using limits just once, proving our formula. Then we can use the formula whenever we need to differentiate a power function.

We begin by using the definition of a derivative.

$\frac{d}{dx}{x}^n=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$

Next, evaluate $f(x+h)$and $f\left(x\right)$.

$\frac{d}{dx}{x}^n=\underset{h\to 0}{\lim }\frac{{(x+h)}^n-x^n}{h}$

We can use The Binomial Theorem to expand ${(x+h)}^n$.

${(x+h)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)x^n+\left(\begin{array}{c}n\\ 1\end{array}\right)x^{n-1}h+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}{h}^2+...+\left(\begin{array}{c}n\\ n\end{array}\right)h^n$

The first two binomial coefficients are 1 and $n$, respectively.

${(x+h)}^n=x^n+n{x}^{n-1}h+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}{h}^2+...+\left(\begin{array}{c}n\\ n\end{array}\right)h^n$

To reflect the definition of a derivative, we need to subtract $x^n$and divide by $h$ on both sides of the equation.

We now have the following expression:

$\frac{{(x+h)}^n-x^n}{h}=n{x}^{n-1}+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}h+...+\left(\begin{array}{c}n\\ n\end{array}\right)h^{n-1}$

As we take the limit as h goes to 0, every term that contains h vanishes. Hence, we are only left with $n{x}^{n-1}$.

Finally, we have arrived at the power rule.

$\frac{d}{dx}{x}^n=n{x}^{n-1}$

Power rule for negative and fractional powers

We only proved the case involving positive integers. However, we can use the power rule when the powers are negative. The formula is the same.

We can also use the power rule for fractional powers, like in the case of a square root function.

More examples of the power rule

Calculus is full of different functions to which we can apply the rules of differentiation. In this section, we will look at more examples of derivatives using the power rule.

The following example considers negative powers.

Let's look at more roots, which we can write as fractional powers.

With enough practice, we can skip some of these steps.

Common mistakes when using the power rule

We cannot use the power rule if the variable is the power of an expression.

Always remember to decrease the power by one after differentiating the function!