Introduction

Welcome to the first article which explains the rules of the derivative. In previous topics, we used the limit definition of the derivative to find the derivative of a function as another function, or at a point. 

In this article, we will learn about a pattern that shows up when we differentiate certain functions. The definition of derivative is still very important, and you should still know it by heart. These shortcuts simply allow us to differentiate functions faster so we can later calculate more complicated ones. 

The Power Rule

Such as in geometry where there are quick and easy formulas that only require you to plug in numbers and simplify, derivatives have their fair share of formulas too. One of these formulas is the power rule. The power rule is something you may see in those introductory “calculus is just simple subtraction and multiplication” Youtube videos. Calculus is a very sophisticated subject, but the power rule will hopefully be easy to understand.

Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1},n\in \mathbb{R}$

The best way to understand what this means is by following an example. 

Using Power Rule

Example 1:

Use the power rule to calculate the derivative of $x^5$ with respect to $x$.

We can simply use the rule and set $n=5$.

$\frac{d}{dx}[x^5]=5x^{(5-1)}=5x^4$

Note that the only actions we took towards differentiating $x^5$ was through plugging in numbers. Let’s try a more complicated example. 

Example 2:

Calculate the derivative of $x^{2/7}$ with respect to $x$.

$\frac{d}{dx}[x^{2/7}]=\frac{2}{7}{x}^{-5/7}$

Notice again that the only actions we did was through plugging in the numbers and following the formula.

Example 3:

Calculate the derivative of $x^{\pi }$ with respect to $x$.

$\frac{d}{dx}[x^{\pi }]=\pi x^{\pi -1}$

You should start to see a pattern to these problems by now. 

There are certain occasions where the power rule does not apply. Notice that for the past three examples, $x$ was raised to a numerical power, whether it was a whole number, a fraction, or an irrational constant. The power rule can only be applied when it is a variable raised to a constant, and cannot be the other way around. 

Example 4:

The derivative of $e^x$ with respect to $x$ is $e^x$. Explain how this doesn’t violate the power rule.

The power rule only works when the variable is just $x$, where $x$ is in the base and the numerical value is the exponent. Therefore as $e^x$ doesn't follow these requirements, the power rule doesn’t apply.

Example 5:

Let $f(x)=\sqrt[5]{x^3}$. Can the power rule be applied here? If so, calculate $f’(x)$. If not, explain why not.

Yes, the power rule can be applied here as $f(x)$ is a function where a variable is raised to a constant power. $f(x)$ can be written as $x^{3/5}.f’(x)=\frac{3}{5}{x}^{-2/5}$. 

Example 6:

Calculate the derivative of $x^2$ with respect to $x$ in two different ways. 

Although there aren’t formal proofs in this class, you should still be prepared for some derivation while still knowing the answer in a second. The two ways we can differentiate the function is by either using the limit definition of the derivative, or using the newly learnt power rule.

Limit Definition of the Derivative:

$\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}$

$=\underset{h\to 0}{\lim }\frac{x^2+2xh+h^2-x^2}{h}$

$=\underset{h\to 0}{\lim }\frac{h(2x+h)}{h}$

$=\underset{h\to 0}{\lim }2x+h=2x$.

Power Rule: $\frac{d}{dx}[x^2]=2x^{(2-1)}=2x$

Practice