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If $ f $ and $ g $ are the functions whose graphs are shown, let $ u(x) = f(g(x)), v(x) = g(f(x)), $ and $ w(x) = g(g(x)). $ Find each derivative, if it exists. If it does not exist, explain why.(a) $ u'(1) $(b) $ v'(1) $(c) $ w'(1) $

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00:19

Frank Lin

04:42

Heather Zimmers

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Catherine A.

October 23, 2020

Heather Z., thanks this was super helpful.

Finally, the answer I needed, thanks Heather Z.

Jessica G.

September 23, 2020

Hi Holly, The chain rule states that the derivative of f(g(x)) is f'(g(x))?g'(x). In other words, it helps us differentiate *composite functions*. Hope that helps.

Holly S.

I'm confused by the term chain rule.

Alexis B.

Hey Sam, In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)).

Sam L.

Can someone explain what the Composite function is?

Frank V.

The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. Hope that helps Hank.

Hank S.

What is a slope?

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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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If $ f $ and $ g $ are the…

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If $f$ and $g$ are the fun…

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Graphs of $f$ and $g$ are …

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If fand g are the function…

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The graphs of $f$ and $z$ …

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Let $f(x)=\ln \left(\frac{…

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that's exercise. We have um a person who is are we have two functions given we want to find the derivative if it exists and if it doesn't exist, Well, explain why. So we have you prime um of one. So based on what we see here, U prime of one is going to be equal to 3/4. So you crime of one. We put the 3/4 and now we want to look at um The prime of one. So the prime of one in this case we see is going to be equal to G prime of two Times F. of one. And this is because of the chain role. So we have a G prime of two kinds of act prime of one. And then lastly we have W Crime one which is going to be G Prime three. Okay, um Time Q prime of one. That's our final answer.

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