Derivative of Volume of Sphere - Sean's Homework

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Homework Help Overview

The discussion revolves around finding the derivative of the volume of a sphere, specifically examining the formula for volume and the implications of substituting the radius with a value derived from circumference.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to differentiate the volume of a sphere using two different expressions for the radius, leading to confusion over obtaining different results. Participants question the application of the chain rule and the consistency of the radius value in different contexts.

Discussion Status

Some participants have offered hints regarding the chain rule, while others express uncertainty about its application and the reasoning behind the differing results. The discussion is ongoing with participants exploring the implications of their approaches.

Contextual Notes

There is a noted confusion regarding the substitution of radius with circumference and how this affects the derivative, highlighting a potential misunderstanding of the relationship between these variables.

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Homework Statement


Find the derivative of volume of a sphere.


Homework Equations


Volume=4/3 * pi * r^3

OR Volume = 4/3 * pi * (c/2pi)^3 where c=circumference

The Attempt at a Solution



This is where I have had some serious trouble. Allow me to explain.

Obviously the derivative is 4pi*r^2

but look what happens when I use r=circumference/2pi

I get a totally different answer. Then the derivative is (c^2)/(2pi^2)

I do not understand why I should get two different answers for the derivative of volume depending only on when I choose to substitute c/2pi for the r value. I would really appreciate it if somebody could help shed some light on what I am doing wrong.

Thanks,

Sean
 
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Hi Sean! :smile:

Hint: chain rule :wink:
 
ok, could you elaborate on that a little. I know what the chain rule is but I don't see it's application here. The result ought to be the same regardless of when I plug in the value of the radius (in this case c/2pi). I don't see why the value of the radius would ever change. That seems like it should remain the same regardless of when i plug it in.
 
Also, where would apply the chain rule here? I don't see it.
 
Oh, I see it now. Thanks.
 
peace :smile:
 

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