Solving Related Rates Problem: Understanding Differentiation and the Power Rule

In summary, The person is questioning why the power rule is not being used on r^2 when differentiating the equation with respect to time. They also ask why only r^2 is being differentiated and not 4 and pi as well. The other person responds that 4pi is a constant and does not need to be differentiated. They also mention that using the product rule would yield the same results.
  • #1
fitz_calc
41
0
Here is the solution to one of my problems:

Picture.jpg


When inserting dV/dt to differentiate the equation as a function of time, why doesn't the book use the power rule on r^2 and multiply the entire equation by 2? I thought when dr/dt was put into the equation you had to differentiate?
 
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  • #2
i don't see why there is a need to multiply by 2? if anything, dividing by 4 would clean things up.

and there is no need to apply the power rule on r^2 b/c it was already differentiated.

sorry i can't give you a more detailed answer.
 
  • #3
Why would they multiply by 2?

they differentiated it correctly.Unless you are differentiating the wrong statement.

[tex]\frac{d}{dt}(\frac{4}{3}\pi r^3)= (3(\frac{4}{3})\pi r^2))\frac{dr}{dt}[/tex]

which is [tex]=4\pi r^2 \frac{dr}{dt}[/tex]
 
  • #4
ahh yeah I was looking at the formula as being 4pi*r^2 instead of cubed, even though it was right in front of me in it's correct form!

Another question given this same example, why do you only differentiate r^2 -- why not 4 and pi as well?
 
  • #5
fitz_calc said:
ahh yeah I was looking at the formula as being 4pi*r^2 instead of cubed, even though it was right in front of me in it's correct form!

Another question given this same example, why do you only differentiate r^2 -- why not 4 and pi as well?
r^3***

and we don't differentiate 4pi bc it's a constant

you could do the product rule on them but you would yield the same results

y=2x
y'=2
 

1. What is a related rates question?

A related rates question is a type of problem in calculus where the rates of change for two or more quantities are related to each other through a given equation. The goal is to find the rate of change of one quantity while the others are changing at a given rate.

2. How do you approach a related rates question?

To solve a related rates question, you need to first identify all the given information and the quantities that are changing. Then, you need to find an equation that relates the changing quantities. After that, you can differentiate the equation with respect to time and solve for the desired rate of change.

3. What are some common examples of related rates questions?

Some common examples of related rates questions include problems involving rates of change of geometric figures (such as the area or volume of a shape), rates of change of distances between moving objects, and rates of change of volume or concentration in a mixing or dilution problem.

4. What are the key concepts involved in solving a related rates question?

The key concepts involved in solving a related rates question are rate of change, derivatives, and the chain rule. It is also important to understand the relationship between the quantities in the problem and how they are changing with respect to time.

5. How can I check my answer for a related rates question?

You can check your answer for a related rates question by verifying that it satisfies all the given information and equations in the problem. You can also check your answer by plugging it back into the original equation and making sure it is a true statement. Additionally, you can use units to make sure your answer has the correct dimensions.

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