Determine the rate of change of the volume of a sphere with respect to its radius.

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Lesson Parent: 
Differentiation V: Derivatives and Rates of Change
Problem: 

Determine the rate of change of the volume of a sphere with respect to its radius.

Answer: 

$4\pi r^2$

SAMPLE SOLUTION

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Solution: 

Recall that the volume of a sphere has the formula

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From this lesson we know that the rate of change of the volume with respect to the radius is $V'(r)$. Now we know that $V(r)$ is a polynomial function and that the derivative of a polynomial is known. Therefore,

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Suppose that we have a sphere of radius <Sign in to see all the formulas>. The volume is

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At a radius of <Sign in to see all the formulas>, $V'(r)$ is

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This is best said that the volume of a sphere will increase <Sign in to see all the formulas> per <Sign in to see all the formulas> increase in its radius when the sphere is at a radius of <Sign in to see all the formulas>.