some fun problems with the product rule

#1

We know how to find the derivative of two functions multiplied together—that’s what the product rule is for. What if we want to find the third derivative of two functions multiplied together? the fourth derivative? fifth? what if we want to find the \(n\)th derivative? \[\begin{align*} \frac{d}{dx}\Big[\, f(x)g(x) \,\Big] &= f'(x)g(x) + f(x)g'(x) \\ \\ \frac{d^2}{dx^2}\Big[\, f(x)g(x) \,\Big] &= \quad ??? \\ \\ \frac{d^3}{dx^3}\Big[\, f(x)g(x) \,\Big] &= \quad ??? \\ \\ \frac{d^4}{dx^4}\Big[\, f(x)g(x) \,\Big] &= \quad ??? \\ \\ \vdots \quad\quad &\quad \quad \vdots\\ \\ \frac{d^n}{dx^n}\Big[\, f(x)g(x) \,\Big] &= \quad ??? \end{align*}\]

#2

We know how to find the derivative of two functions multiplied together—that’s what the product rule is for. But what if we have three functions multiplied together? what if we have four functions multiplied together? five? what if we have \(n\) functions multiplied together??? (For notational convenience as you try to figure this out, you may wish to write your functions as \(f_1(x)\), \(f_2(x)\), \(f_3(x)\), and so on, because there are only 26 letters. And you might wish to drop the ``\((x)\)’’, too, since it creates a lot of redundant writing.) \[\begin{align*} \frac{d}{dx}\Big[\, f_1f_2 \,\Big] &= f_1'f_2 + f_1f_2' \\ \\ \frac{d}{dx}\Big[\, f_1f_2f_3 \,\Big] &= \quad ??? \\ \\ \frac{d}{dx}\Big[\, f_1f_2f_3f_4 \,\Big] &= \quad ??? \\ \\ \vdots \quad\quad &\quad \quad \vdots\\ \\ \frac{d}{dx}\Big[\, f_1f_2f_3\cdots f_n \,\Big] &= \quad ??? \end{align*}\]

#3

What about… the \(n\)th derivative of \(m\) functions multiplied together?!?