higher derivatives

Question

Use the product rule three times to find a formula for (fg)''' and compare the result with the expansion (a+b)³. Then try to guess a general formula for (fg)⁽ⁿ⁾.

I don't really understand what they are asking me here and it would be really helpful if someone could explain it to me.

Answer 1

Let's break this down.

Use the product rule three times to find a formula for $(fg)''' \ldots$

This means to compute the third derivative of the function $h(x) = f(x)g(x)$. That is, find the derivative $h',$ then differentiate that again, and then differentiate the result. (Cf. second derivative.)

...and compare the result with the expansion $(a+b)^3.$

You can just do the ordinary FOIL technique, or, if you know the binomial theorem (which is a big plus here), use it. See what the two expansions have in common.

Then try to guess a general formula for $(fg)^{(n)}.$

Can you find the pattern and find a formula for an $n$-th derivative of a product?

These are all the hints I'll give.

Answer 2

We apply the product (and sum) rules on derivatives: $$\matrix{(fg)'&=&f'g+fg' \\ (fg)''&=&(f'g+fg')'&=&(f'g)' +(fg')'=&\dots \\ (fg)'''&=&\dots}$$