If $ F(x) = f(xf(xf(x))), $ where $ f(1) = 2, f'(2) = 3, f'(1) = 4, f'(2) = 5, $ and $ f'(3) = 6, $ find $ F'(1). $
Okay, so here we have f of X and we want to find f prime of X. And let me just say that whoever was writing this problem, I think was having a little bit too much fun. And if someone actually assigned it to you to do I just want to say I am sorry. So we're gonna work our way through this the best we can. But we have here is the product rule inside the chain rule inside the product rule inside the chain rule. So so lot of efs and a lot of exes, and it's going to look pretty complicated. Okay, so we're starting with the derivative of the outside function. The outside function is F so we have f prime of everything inside it. So if f prime of x f of x f of X, Okay, now we multiply that by the derivative of the inside, the inside is a product X times the other part. So we need the product rule. So we have the first X times, the derivative of the second. While here comes the next implementation of the chain rule. Because the derivative of the second is going to be the derivative of this f of x times f of X. Okay, so that will be f prime of its inside times, the derivative of its inside. So the derivative of X f of X well, that is a product. So we need to use the product rule on that. So that would be the first x times, the derivative of the second of prime of X plus the second f of x times, the derivative of the 1st 1 That part wasn't too bad, but let's take stock of where we are. This was the first times the derivative of the second in our product rule we still have to do. Plus, the second time's a derivative of the first and the second is all of this f of x times f of X to the second times. The derivative of the first, the first was X. It's derivative is one who that's not bad. Okay, that's it. That's the whole derivative. Now we're plugging in one, and we have some values we were given. So f prime of one is now. Here's where we pay very close attention to detail. We have f prime of one times f of one times f of one times. I don't know if the color coding will help, but I'll try it one times f prime of one times f of one times one times f prime of one plus f of one plus f of one times f of one. Not too bad. Now, every time we have an f of one, we substitute Where did it go? F one is to So every time we have enough of one, we substitute to and have prime of one is for Okay, so we can do that. So we have f prime of now. I'm not gonna write this one that I wrote here. OK, we don't need one time something to of f prime of f of one times f of one So that would be one times to This is to times now we're moving on to read. We had one times f prime of one times f of one. So here's another f of one. So here's another two times now in the green, we had one times f prime of one, so that f prime of one is for plus f of one that f of one is to. Plus, we had f of one times f of one, and that f of one is to Okay, we're getting there now. Every time we see enough of to, we're going to substitute a three. And every time we see an F prime of two, we're going to substitute of five. So we have f prime of three times five times four plus 26 plus f of to three. Okay, it's getting smaller and smaller now. F prime of three is six, so we're going to substitute a six for F Prime of three. So if six times, five times six is 30 plus three is 33 six times 33 is 1 98 That is our answer.