Concept preview

What you will be able to do

1. 01State the power rule $\frac{d}{dx}x^n = n x^{n-1}$ and explain why it holds for every real exponent — positive, negative, or fractional.
2. 02Derive the power rule from the limit definition for $x^2$ and $x^3$ using binomial expansion and cancellation of $h$.
3. 03Rewrite roots and reciprocals as powers first, then differentiate and evaluate the derivative at a point.

Here is the whole rule in one line. For any real number $n$, $\frac{d}{dx}\,x^n = n\,x^{\,n-1}.$ In words: bring the exponent down in front as a coefficient, then subtract one from the exponent. Applied to $x^5$ you get $5x^4$; applied to $x^{10}$ you get $10x^9$. The two operations always happen together — the old exponent becomes the new coefficient, and the new exponent is one smaller. Miss either step and the answer is wrong, so it is worth saying the rule aloud as you use it: “down in front, minus one.”