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

 Sep 26, 2016

Best Answer 

 #10
avatar+33603 
+5

More generally:

 

.

 Sep 30, 2016
 #1
avatar+118587 
+5

This is a really unusual question.

 

Let r(x)=F(g(h))),

 

where h(1)=2,        g(2)=5,        h'(1)=4,          g'(2)=4,          and      f'(5)=6.     [I will assume tht f'(5) is really F'(5)]

Find r'(1).

 

\(r(x)=F(g(h))\\ r(x)=F(g(h(x)))\\ r'(x)=F'(g(h(x)))\\ r'(1)=F'(g(h(1)))\\ r'(1)=F'(g(2))\\ r'(1)=F'(5)\\ r'(1)=6\\ \)

 Sep 26, 2016
 #2
avatar+33603 
+5

Hmm!

 

.

 Sep 26, 2016
 #3
avatar+118587 
+5

I stand corrected.

 

Thanks Alan :))

 Sep 26, 2016
 #4
avatar+118587 
+5

So does this mean that the answer cannot be determined without more information?

Melody  Sep 26, 2016
 #5
avatar+33603 
+5

The answer is unique because:

 

r`(x) = f`(g(h(x)))*g`(h(x))*h`(x)

 

.

Alan  Sep 26, 2016
 #6
avatar+118587 
0

Ok - I better think about it better...   Thanks Alan :D

Melody  Sep 26, 2016
 #7
avatar+118587 
+5

Hi Alan,

I have been trying to work out what you have said.

I understand your example of  course but I do not understand why that means that this is the only answer.  How can you take just one example and say the answer is always going to be the same.??

 

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

 

I think this is right.   

 

\(\begin{align} \frac{dr}{dx}&=\frac{df}{dg} \cdot \frac{dg}{dh}\cdot \frac{dh}{dx}\\ \end{align} \)

 

 I'm told that  when x=1     dh/dx =2   but I don't know from the info given what the other two derivatives are.  How can I use all that other information without resorting to a specific example ???

 Sep 29, 2016
 #8
avatar+33603 
+5

Here's another example with completely different functions, but with the same values of f`(5), g(2) etc.:

 

Still get 96.  More than a coincidence!

Alan  Sep 30, 2016
 #9
avatar+118587 
0

Thanks Alan,

I never doubted you but I'd still like to see it done in a more general way. ://

Melody  Sep 30, 2016
 #10
avatar+33603 
+5
Best Answer

More generally:

 

.

Alan Sep 30, 2016
 #11
avatar+118587 
0

Thanks Alan,

I shall take a very good look at your answer :)

Melody  Sep 30, 2016

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