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Let $ r(x) = f (g(h(x))), $ where $ h(1) = 2, g(2) = 3, h'(1) = 4, g'(2) = 5, $ and $ f'(3) = 6. $ Find $ r'(1). $

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01:09

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Oregon State University

University of Nottingham

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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01:54

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our goal is to find our prime of one. We have R of X. It's a composite function. So we're going to use the chain rule and let's start by finding our prime of X. So we start with the derivative of the outside function F with F prime of G of h of X. Then we moved to the middle function G, and we have its derivative g prime of h of X. Then we moved to the inside function H. And it's derivative is age prime of X. Now we want to substitute one in here so our prime of one would be f prime of G of age of one times g, prime of age of one times h, prime of one. Now we can start substituting some of the numbers in here. H of one is to we get that again and then we get h prime of one is for So when we substitute those in, we have f prime of G of to times G prime of to times four. Now we need G of to G of two is three and when you g prime of two g prime of two is five So now we have f prime of three times. Five times four. Okay, One more step to go. We need F prime of three f prime of three of six. So now we have six times, five times four and that's 1 20

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