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

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01:20

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Oregon State University

Harvey Mudd College

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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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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