Differentiation of the product of any finite number of functions

Now in dividing both sides of equation (V) by $uv$, this formula assumes the form

$$\frac{\frac{d}{dx}(uv)}{uv} = \frac{\frac{du}{dx}}{u} + \frac{\frac{dv}{dx}}{v}.$$

If then we have the product of $n$ functions $y = v_1 v_2 \cdots v_n$, we may write

$$\begin{array}{ll} \frac{\frac{d}{dx}(v_1 v_2 \cdots v_n)}{v_... ...cdots + (v_1 v_2 \cdots v_{n - 1})\frac{dv_n}{dx}. \end{array}$$

The derivative of the product of a finite number of functions is equal to the sum of all the products that can be formed by multiplying the derivative of each function by all the other functions.