Related rate problems and the chain rule

In summary, the conversation discusses the use of the chain rule in related rates problems. The person is able to follow the procedure outlined in the text but wants to understand why it works. They mention a specific example involving a balloon and the volume increasing at a constant rate. The confusion lies in applying the chain rule to the formula V=(4*pi*r^3)/3. They mention the general form of the chain rule and how it works, but are unsure of how it applies to related rates problems. Another person provides an explanation and clarifies that the chain rule is being applied to V(t)=F(r(t)), where F(r) is the formula for volume.
  • #1
kdinser
337
2
I'm having some trouble understanding why the chain rule comes into play in related rates problems. I'm able to follow the procedure outlined in the text and come up with the correct answer, but I'm not really comfortable moving on from a section until I understand why it works.

The standard first example of this problem in the 3 books that I have deals with a balloon blowing up at a constant rate and the rate that the radius of the balloon is changing over time as the volume increases at that rate.

I'm fine with everything in this problem except the part where they take the derivative of both sides and apply the chain rule to this formula.

V=(4*pi*r^3)/3

Actually using the chain rule, I'm fine with and I've gone through enough proofs to understand how and why it works.
h(x)=f(g(x))
h'(x)=f'(g(x))g'(x)

What I'm not getting is, how does it work with related rates problems. When you take the derivative of both sides of V=(4*pi*r^3)/3, where is the f(g(x)) that the chain rule is being applied to?
 
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  • #2
V(t)=F(r(t)), F(r)=4/3*pi*r^(3)
 
  • #3
Thanks arildno, it makes sense now.
 

1. What are related rate problems?

Related rate problems are mathematical problems that involve finding the rate of change of one variable with respect to another variable. They typically involve multiple variables that are related to each other through a given equation or situation.

2. How do you solve related rate problems?

To solve related rate problems, you first need to identify the variables involved and how they are related to each other. Then, you can use the chain rule, which is a mathematical rule for finding the derivative of a composite function, to find the rate of change of the desired variable.

3. What is the chain rule?

The chain rule is a mathematical rule that allows you to find the derivative of a composite function by breaking it down into simpler functions and finding the derivatives of each individual function. It is used in related rate problems to find the rate of change of one variable with respect to another variable.

4. What are some real-life applications of related rate problems?

Related rate problems are commonly used in fields such as physics, engineering, and economics to analyze and predict changes in various systems. For example, they can be used to find the rate at which the volume of a balloon is changing as it is being filled with air, or the rate at which the cost of a product is changing as the demand for it changes.

5. How can I improve my problem-solving skills for related rate problems?

To improve your problem-solving skills for related rate problems, it is important to have a strong understanding of the chain rule and how to apply it. You can also practice solving various related rate problems and familiarize yourself with different types of scenarios that may require the use of this rule.

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