Determine the rate of change of the circumference of a circle with respect to its radius.

Primary tabs

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

Determine the rate of change of the circumference of a circle with respect to its radius.

Answer: 

$2\pi$

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Recall that the circumference of a circle has the formula

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>


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

<Sign in to see all the formulas>


This is best said that the circumference of a circle will increase <Sign in to see all the formulas> per <Sign in to see all the formulas> increase in its radius.