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If $ F(x) = f(3f(4f(x))), $ where $ f(0) = 0 $ and $ f' (0) = 2, $ find $ F' (0). $

$F^{\prime}(0)=96$

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here we have capital f of X. It's a composite function. It has three layers. We've got the F on the outside, We have the three F in the middle and then we have the four F on the very inside and our goal is to find capital F prime of zero. So we have to start by finding capital f prime of X. So using the chain rule, we start with the derivative of the outermost function. So we have f prime of its inside three f of four f of X. Now we multiply by the derivative of the middle function, the three F function. So the derivative of three F would be three times f prime. So we have three times f prime of its inside for f of X. Now we need to multiply by the derivative of the inside function and the derivative of four f of X would be four f prime. Okay, Now, we're not gonna worry about simplifying that right now because our goal is to find the value at zero. So let's go ahead and substitute a zero in there at this point. Sweet F prime of three f of four f of zero times three f prime of four F of zero times four of prime a zero. Now we can take the values that were given. F of zero is zero weaken substitute that in in two places and f prime of zero is too. And we can substitute that in one place. And now we have f prime of three F of four times zero times three f Prime of four times zero times four times two. Okay, let's clean that up a little bit. It's we have f prime of three F of zero times three of prime of zero times eight. Now we can go ahead and find f of zero again and substitute that End that zero and f prime of zero. Substitute that, and that's going to be too. So we have f prime of three times zero times three f Prime of zeer Looks that when we know that one was to prime of zero was too. Times eight. Okay, so we have f prime of zero times three times two times eight and that would be times 48. Now, one more time. F prime of zero is too. So with two times 48 so we get 96

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