The Product Rule

The derivative of a product isn't the product of the derivatives. For $x \cdot x = x^2$, the rule gives $1 \cdot x + x \cdot 1 = 2x$, which is right, while $f'g'$ would give $1$.

To prove the rule, add and subtract $f(x + h)g(x)$ in the numerator of the difference quotient:

As $h \to 0$, the two quotients approach $g'(x)$ and $f'(x)$. The factor $f(x + h)$ approaches $f(x)$, because a differentiable function is continuous. The limit is $f(x)g'(x) + g(x)f'(x)$.

In the picture, the proof's $f(x + h)\Delta g$ is the top strip together with the corner, and $g(x)\Delta f$ is the right strip. Divide the added area by $h$. The strips give $g \cdot \frac{\Delta f}{h} \to gf'$ and $f \cdot \frac{\Delta g}{h} \to fg'$. The corner gives $\frac{\Delta f}{h} \cdot \Delta g \to f' \cdot 0 = 0$, since $\Delta g \to 0$ when $g$ is continuous.