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This is the topic of: Derivatives of Composite functions

This is suppose to help me discover a rule but I just got confused.

Here goes:

Suppose y = u2

a. Find dy/du                                                            (My answer was "2u")

b. Now suppose u = ax + 1, so y = (ax + 1)2

i. Find du/dx                                                             (My answer was "a", feel like it's wrong)

ii. Write dy/du from a in terms of x                           (My answer was "2x")

iii. Hence find dy/du * du/dx                                     ("2ax"? Right?)

iv. Compare your answer to question 3.                  (It was 2a2 + 2a, last time I checked "2ax" =/= "2a2 + 2a")

 

I have difficulties understanding what

d(something)/d(something) means.

Like if it says dy/dx, I would understand it to be to find the derivative of y. So if it says, dy/du, does it mean to find the derivative of y? But it's compared to "du" instead of "dx". Sorry if you can't understand what I'm trying to say, hard to put it into words my confusion. 

 

My answers are obviously wrong, anyone care to help and correct it. That way maybe I might understand.

Thanks.

Guest Jul 25, 2017
edited by Guest  Jul 25, 2017

Best Answer 

 #1
avatar+26357 
+2

dy/dx means differentiate y with respect to x

dy/du means differentiate y with respect to u

du/dx means differentiate u with respect to x

 

y = u2        dy/du = 2u

 

u = ax + 1  du/dx = a

 

Since dy/du = 2u and u = ax + 1 we must have dy/du = 2(ax + 1) by simply replacing u on the right hand side by ax + 1.

 

So dy/du*du/dx = 2(ax + 1)*a → 2a(ax + 1) or 2a2x + 2a

 

Notice that dy/dx = dy/du*du/dx (this is known as the chain rule) so dy/dx = 2a2x + 2a.   

 

(See http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-chain-2009-1.pdf for more information on the chain rule.)

 

Hope this helps!

Alan  Jul 25, 2017
edited by Alan  Jul 25, 2017
edited by Alan  Jul 25, 2017
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1+0 Answers

 #1
avatar+26357 
+2
Best Answer

dy/dx means differentiate y with respect to x

dy/du means differentiate y with respect to u

du/dx means differentiate u with respect to x

 

y = u2        dy/du = 2u

 

u = ax + 1  du/dx = a

 

Since dy/du = 2u and u = ax + 1 we must have dy/du = 2(ax + 1) by simply replacing u on the right hand side by ax + 1.

 

So dy/du*du/dx = 2(ax + 1)*a → 2a(ax + 1) or 2a2x + 2a

 

Notice that dy/dx = dy/du*du/dx (this is known as the chain rule) so dy/dx = 2a2x + 2a.   

 

(See http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-chain-2009-1.pdf for more information on the chain rule.)

 

Hope this helps!

Alan  Jul 25, 2017
edited by Alan  Jul 25, 2017
edited by Alan  Jul 25, 2017

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