# Power rule review

Review your knowledge of the Power rule for derivatives and solve problems with it.

## What is the Power rule?

The Power rule tells us how to differentiate expressions of the form $x^n$ (in other words, expressions with $x$ raised to any power):
$\dfrac{d}{dx}x^n=n\cdot x^{n-1}$
Basically, you take the power and multiply it by the expression, then you reduce the power by $1$.

## Differentiating polynomials

The Power rule, along with the more basic differentiation rules, allows us to differentiate any polynomial. Consider, for example, the monomial $3x^7$. We can differentiate it as follows:
\begin{aligned} \dfrac{d}{dx}[3x^7]&=3\dfrac{d}{dx}(x^7)\quad\gray{\text{Constant multiple rule}} \\\\ &=3(7x^6)\quad\gray{\text{Power rule}} \\\\ &=21x^6 \end{aligned}
Problem 1
$f(x)=x^5+2x^3-x^2$
$f'(x)=$

Want to try more problems like this? Check out this exercise.

## Differentiating negative powers

The Power rule also allows us to differentiate expressions like $\dfrac{1}{x^2}$, which is basically $x$ raised to a negative power. Consider this differentiation of $\dfrac{1}{x^2}$:
\begin{aligned} \dfrac{d}{dx}\left(\dfrac{1}{x^2}\right)&=\dfrac{d}{dx}(x^{-2})\quad\gray{\text{Rewrite as power}} \\\\ &=-2\cdot x^{-3}\quad\gray{\text{Power rule}} \\\\ &=-\dfrac{2}{x^3}\quad\gray{\text{Rewrite as fraction}} \end{aligned}
Problem 1
$\dfrac{d}{dx}\left(\dfrac{-2}{x^4}+\dfrac{1}{x^3}-x\right)=$

Want to try more problems like this? Check out this exercise.

## Differentiating fractional powers and radicals

The Power rule also allows us to differentiate expressions like $\sqrt x$ or $x^{^{\frac{2}{3}}}$. Consider this differentiation of $\sqrt x$:
\begin{aligned} \dfrac{d}{dx}\sqrt x&=\dfrac{d}{dx}\left(x^{^{\Large\frac{1}{2}}}\right)\quad\gray{\text{Rewrite as power}} \\\\ &=\dfrac{1}{2}\cdot x^{^{\Large-\frac{1}{2}}}\quad\gray{\text{Power rule}} \\\\ &=\dfrac{1}{2\sqrt x}\quad\gray{\text{Rewrite as radical}} \end{aligned}
Problem 1
$f(x)=6x^{^{\Large\frac{2}{3}}}$
$f'(x)=$

Want to try more problems like this? Check out these exercises: