Power Rule, Part 2

In part 1, we saw that the power rule $\frac{d}{dx} x^n = nx^{n-1}$ works in the following cases:

Next we'll figure out whether it works in other cases.

Negative integer

Let $n$ be any negative integer. Because $n$ is negative, we have $n = (-1)\abs{n}$, and so $$ x^n = x^{(-1)\abs{n}} = \left( x^{\abs{n}} \right)^{-1}. $$ We know that the power rule works in the case $n = -1$, so $\frac{d}{dx} x^{-1} = (-1)x^{-2}$. By using that with the chain rule, we get $$ \frac{d}{dx} x^n = \left( x^{\abs{n}} \right)^{-1} = (-1)\left( x^{\abs{n}} \right)^{-2} \cdot \frac{d}{dx} x^{\abs{n}}. $$ By using the power rule with the positive integer $\abs{n}$ (we know that it works for positive integers), we get $$ \frac{d}{dx} x^{\abs{n}} = \abs{n}x^{\abs{n} - 1}. $$ By bringing it all together, we get $$ \begin{align} \frac{d}{dx} x^n &= (-1)\left( x^{\abs{n}} \right)^{-2} \abs{n}x^{\abs{n} - 1} \\ &= -\abs{n}x^{(-2)\abs{n}}x^{\abs{n} - 1}. \end{align} $$ Because $n$ is negative, we have $n = -\abs{n}$ and $\abs{n} = -n$, so $$ \begin{align} \frac{d}{dx} x^n = nx^{2n}x^{-n - 1} = nx^{2n-n-1} = nx^{n-1}. \end{align} $$

The power rule works when $n$ is a negative integer.

Rational number

Let $a$ be any integer, and let $b$ be any positive integer. Now $$ x^{a/b} = \left( x^a \right)^{1/b}. $$ We know that the power rule works when its $n$ is $1/b$ or any integer, so $$ \frac{d}{dx} x^{1/b} = \frac 1 b x^{1/b\ -1}, \qquad \frac{d}{dx} x^a = ax^{a-1}. $$ The chain rule gives $$ \begin{align} \frac{d}{dx} x^{a/b} &= \frac{d}{dx} \left( x^a \right)^{1/b} \\ &= \frac{1}{b} \left( x^a \right)^{1/b\ -1} ax^{a-1} \\ &= \frac{a}{b} x^{a ( 1/b\ -1 )} x^{a-1} \\ &= \frac{a}{b} x^{a ( 1/b\ -1 ) + (a-1)}. \end{align} $$ Simplifying the power of $x$ gives $$ a \left( \frac 1 b - 1 \right) + (a-1) = a\frac 1 b - a + a - 1 = \frac a b - 1, $$ so we get $$ \frac{d}{dx} x^{a/b} = \frac a b x^{a/b\ -1}. $$

The power rule works when $n=a/b$, where $a$ is any integer and $b$ is a positive integer.

Any fraction of integers $\frac{A}{B}$ can be written as an integer divided by a positive integer. First of all, $B$ can't be zero, because we can't divide by zero. If $B$ is negative, then we have $$ \frac{A}{B} = \frac{-A}{-B}, $$ where $-A$ is an integer and $-B = \abs{B}$ is a positive integer.

The power rule works when $n$ is a fraction of integers.

This result includes all of our previous power rule results, including the results for integers; any integer $k$ can be written as a fraction of integers, because $k = \frac k 1$.

A number is called rational, if it can be written as a fraction of integers, so the above result could be also written as follows: the power rule works when $n$ is rational.

Not all numbers are rational. For example, $\sqrt{2}$ is not rational (find a proof from your favorite math YouTube channel). Those numbers are called irrational, and later (TODO) we'll figure out what happens to the power rule when $n$ is irrational.