The power, product and quotient rules
Rules from rectangles
If you know how fast f changes and how fast g changes, how fast does the product f\cdot g change? Is it just f'\cdot g'?
Picture f(x)\,g(x) as the area of a rectangle with width f(x) and height g(x). Nudge x by \Delta x and both sides grow. Where does the new area come from?
Indigo outline: the original rectangle. Ochre strip: g · Δf. Indigo strip: f · Δg. The dark corner is Δf · Δg.
The rules
Dividing the new area by \Delta x gives three pieces. As \Delta x\to0, the two strips approach g\,f' and f\,g', while the corner, which is small in both directions, vanishes. That is the product rule:
The other basic rules come from similar arguments. Together they let you differentiate any polynomial or rational function without taking a limit.
| Rule | Statement |
|---|---|
| Constant | \frac{d}{dx}c=0 |
| Power | \frac{d}{dx}x^n=nx^{n-1} for any real n |
| Constant multiple and sum | (cf)'=cf',\quad (f\pm g)'=f'\pm g' |
| Product | (fg)'=f'g+fg' |
| Quotient | \left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2} |
Why does the power rule work for n=3? Expand (x+h)^3=x^3+3x^2h+3xh^2+h^3. Subtract x^3, divide by h, and let h\to0. Only 3x^2 survives. The same pattern holds for every positive integer n.
Rewrite roots and reciprocals as powers before differentiating: \sqrt{x}=x^{1/2} and \tfrac{1}{x^3}=x^{-3}.
Solve it step by step
Choose the rule, find the derivative, then evaluate it. Answers can include pi, as in pi/2. The score counts steps you answer correctly on the first try.
Common mistakes
- Multiplying the derivatives
(fg)'\ne f'g'. For x\cdot x, that would give 1, but the derivative of x^2 is 2x. The rectangle shows the two strips you would be missing.
- Reversing the quotient rule
The order in the numerator matters: f'g-fg', with the derivative of the top first.
- Using the power rule on a sum in a denominator
\tfrac{1}{x+1} is not x^{-1}+1. Use the quotient rule, or the chain rule from the next lesson.