Why does volume's derivative differ when substituting circumference for radius?

[peacemaster]

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

 

[tiny-tim]

Hi Sean!

Hint: chain rule

 

[peacemaster]

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.

 

[peacemaster]

Also, where would apply the chain rule here? I don't see it.

 

[peacemaster]

Oh, I see it now. Thanks.

 

[tiny-tim]

peace