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Let $ f $ and $ g $ be the function in Exercise 63.(a) If $ F(x) = f(f(x)), $ find $ F'(2). $(b) If $ G(x) = g(g(x)), $ find $ G'(3). $
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00:57
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 4
The Chain Rule
Derivatives
Differentiation
Missouri State University
Campbell University
Harvey Mudd College
Idaho State University
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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in this problem. Capital F of X is f of f of X and we want to find capital f prime of to. So let's find capital f prime of X using the chain rule The derivative of the outside would be f prime of f of x times, the derivative of the inside F prime of X. Now let's evaluate that at two. So capital F prime of two would be f prime of f of to times f prime of to Now we use the table to find f of to f of two is one and f prime of to if prime of two is five and we substitute those numbers in So we have f prime of one times five. Now we go back to the table to find f prime of one of prime of oneness four Suite four times five. So the answer is 20. And for part B, we have capital G FX, which is G of G of X and we want to find capital G prime of three. So let's find capital G prime of X using the chain rule. The derivative of the outside would be g prime of G of X Times, the derivative of the inside G prime of X. Now let's evaluate that at three. So G prime of three would be g prime of G of three times g Prime of three. Now let's use the table to find G of three. G of three is too. And to find g prime of three g Prime of three is nine so we can substitute those values in and we get G prime of to times nine. Now we can look back at the table for G Prime of two g Prime of two is seven to have seven times nine, so the answer is 63.
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