Derivative Power Rule
Simple overview of the derivative power rule.
Main Idea
We use the derivative power rule when taking the derivative of functions containing powers.
The main structure of the simplest application of the power rule is
\[\frac{d}{dx}\left[x^n\right]=nx^{n-1}\]Here when differentiating $x^n$ we bring the power out front and reduce the power by $1$ to get \(\frac{d}{dx}\left[x^n\right]=nx^{n-1}\).
Watch this section: introduction at 0:00.
Example 1
Consider
\[f(x)=x^3\]Bring the $3$ down in front:
\[\frac{d}{dx}\left[x^3\right]=3x^{3-1}\]Then subtract $1$ from the exponent:
\[3x^{3-1}=3x^2\]So the derivative is
\[f'(x)=3x^2\]Watch this section: example 1 at 0:40.
Constant Multiple Example
Consider the following
\[f(x)=5x^7\]The coefficient $5$ stays in front while the power rule handles $x^7$:
\[\frac{d}{dx}\left[5x^7\right] = 5\frac{d}{dx}\left[x^7\right]\]Now bring down the $7$ and subtract $1$ from the exponent:
\[5\frac{d}{dx}\left[x^7\right] = 5\left(7x^{7-1}\right)\]Then multiply the constants:
\[5\left(7x^6\right)=35x^6\]So
\[f'(x)=35x^6\]Watch this section: constant multiple example at 1:00.
Example 2
Consider
\[f(x)=2x^5+x^3+x\]Differentiate one term at a time
\[\frac{d}{dx}\left[2x^5\right]=10x^4\] \[\frac{d}{dx}\left[x^3\right]=3x^2\]and
\[\frac{d}{dx}\left[x\right]=1\]Putting the pieces together gives
\[f'(x)=\frac{d}{dx} \left[2x^5+x^3+x\right] =\frac{d}{dx}\left[2x^5\right] + \frac{d}{dx}\left[x^3\right] + \frac{d}{dx}\left[x\right] =10x^4+3x^2+1\]Watch this section: example 2 at 1:29.
Timestamp Guide
More Practice
For timestamped examples from this video and related derivative lessons, use the Differentiation Power Rule Example Archive.
You can also review the derivatives topic page, compare this lesson with the product rule companion, or continue to the chain rule companion.
The video description points to JoeCMath’s derivative rules playlist: watch the playlist on YouTube.