Derivative of Volume of Sphere - Sean's Homework

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In summary, the conversation discusses finding the derivative of volume for a sphere using two different equations, one using the radius and the other using the circumference. The difference in the results can be attributed to the application of the chain rule, where the derivative of the volume equation using the circumference as the radius value gives a different result due to the chain rule.
  • #1
peacemaster
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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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  • #2
Hi Sean! :smile:

Hint: chain rule :wink:
 
  • #3
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.
 
  • #4
Also, where would apply the chain rule here? I don't see it.
 
  • #5
Oh, I see it now. Thanks.
 
  • #6
peace :smile:
 

1. What is the formula for finding the derivative of the volume of a sphere?

The formula for finding the derivative of the volume of a sphere is dV/dR = 4πR^2, where dV is the derivative of the volume and R is the radius of the sphere.

2. Why is it important to know the derivative of the volume of a sphere?

Knowing the derivative of the volume of a sphere is important because it allows us to calculate the rate of change of the volume with respect to the radius. This can be useful in various applications, such as in physics, engineering, and economics.

3. How is the derivative of the volume of a sphere related to its surface area?

The derivative of the volume of a sphere is related to its surface area through the formula dV/dR = 4πR^2, where dV is the derivative of the volume and R is the radius. This formula shows that the derivative of the volume is directly proportional to the surface area of the sphere.

4. Can the derivative of the volume of a sphere be negative?

Yes, the derivative of the volume of a sphere can be negative. This would indicate that the volume is decreasing as the radius increases, or that the sphere is shrinking.

5. How can the derivative of the volume of a sphere be used in real life scenarios?

The derivative of the volume of a sphere can be used in real life scenarios, such as in calculating the rate of change of a balloon's volume as it is being inflated or deflated. It can also be used in determining the optimal size of a container for a given volume, as well as in predicting the growth of biological cells or organisms.

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