# A calculus of the absurd

#### 11.4 The product rule

Proving this is a little tricky, and needs some ingenuity. The product rule gives us a way to find the derivative of a function which is the product of two functions $$f(x) = a(x) \cdot b(x)$$.

The trick here is to "add zero"

If (as it does) $$h \to 0$$ then $$a(x + h) \to a(x)$$, we can rewrite the limit as

Overall, we therefore can say that the derivative of a function $$f(x) = a(x)b(x)$$ is

$$\frac {df}{dx} = \frac {d}{dx}[a(x)]b(x) + a(x)\frac {d}{dx}[b(x]$$