The Knowledge Web

The Product Rule

Imagine a rectangle where the width is $f(x)$ and the height is $g(x)$ . As $x$ changes, both sides grow.

The Geometric Proof

Area = u · v

Why isn't it just f'g'?

When both sides of a rectangle grow, you get three new areas.
However, the white corner piece ( $du \cdot dv$ ) is so tiny compared to the strips that it vanishes to zero.

Change in Area

d(uv) = u \cdot dv + v \cdot du

Change in u (du)1.0

Change in v (dv)0.8

This geometric logic gives us the rule: "Left times derivative of Right, plus Right times derivative of Left."

\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)

The Quotient Rule

Division is just messy multiplication. The rule for $f(x) / g(x)$ is infamous for being hard to memorize.

The Formula

\frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}

The Song (Mnemonic)

"Low D-High minus High D-Low,
Draw the line and down below,
Square the bottom and off we go!"