Derivatives Warmup & Power Rule

Differentiate c x^n one piece at a time.

The idea

The derivative measures how fast a function is changing at a single instant.

For a straight line the rate of change is its slope: take two points, divide the rise by the run, and one number describes the whole line. A curve has no such number, because it bends — the slope through two of its points is only an average rate across that stretch, and it changes when either point moves.

So hold one point fixed at $x$ and place the second a short distance $h$ along, at $x + h$. The line through them has slope $\dfrac{f(x + h) - f(x)}{h}$, the change in output divided by the change in input. Now shrink $h$: ten times smaller, then a thousand times smaller, and so on. The averages settle on one number.

That number is the derivative.

Definition (Derivative).

The derivative of $f$ at $x$, written $f'(x)$ and read as f prime of x, is $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h},$ provided this limit exists.

As $h$ shrinks, the line through the two points tilts into the tangent line to the graph at $x$; its slope is $f'(x)$, the rate at that one point.

Ways to work on it

• Walkthrough. Derive the power rule from the limit definition, then apply it.
• Practice. Differentiate a random cx^n with the power rule.
• Hardest. One step. Two-term polynomial — differentiate and combine in your head.

Not sure where to start? Take the ten-question placement test.