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Problem 65 Hard Difficulty

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) $

Answer

a. $\frac{3}{4}$
b. $g^{\prime}(2) f^{\prime}(1)$
c. $g^{\prime}(3) g^{\prime}(1)$

Discussion

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hS

Hank S.

September 23, 2020

What is a slope?

FV

Frank V.

September 23, 2020

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.

Sl

Sam L.

September 23, 2020

Can someone explain what the Composite function is?

AB

Alexis B.

September 23, 2020

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)).

HS

Holly S.

September 23, 2020

I'm confused by the term chain rule.

JG

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.

CA

Catherine A.

October 23, 2020

Finally, the answer I needed, thanks Heather Z.

CA

Catherine A.

October 23, 2020

Heather Z., thanks this was super helpful.

Video Transcript

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.