Basic differentiation?

In summary: This way, you can find the derivative of V with respect to t.Hope that helps!In summary, the conversation is about a problem with the chain rule in a calculus book. The person is having trouble understanding why the rule is being used and where the terms in the equation come from. They are confused about the basic concepts and thank the other person for their help. The expert explains that the chain rule is necessary because the variable r is a function of t and the rule allows them to find the derivative of V with respect to t.
  • #1
moooocow
12
0
Ok, In my calc book I am having problem with the first example they give in a chapter on Differentiation.

It gives the equation of [itex]V = \frac{4}{3} \pi r^3[/itex] and then it is differentiated by time = t.

[tex]\frac{dV}{dt} = \frac {dV}{dr} \frac{dr}{dt} = 4 \pi r^2 \frac {dr}{dt}[/tex]

I don't quite understand why the chain rule is being used in the first place, because [itex]4 \pi r^2[/itex] doesn't need it, and also I don't understand where [itex]\frac{dr}{dt}[/itex]comes from in the last part. I am quite confused by this basic stuff [b(] so thanks for any help.

edit : \/ Thanks for the help getting it to show up right :)
 
Last edited:
Physics news on Phys.org
  • #2
Originally posted by moooocow
Ok, In my calc book I am having problem with the first example they give in a chapter on Differentiation.


It gives the equation of [itex]V = \frac{4}{3} \xi r^3[/itex] and differentiates each side in respect to time = [itex]t[/itex]. It gets the equation:

[tex]\frac{dV}{dt} = \frac {dr}{dt} \frac{dr}{dt} = 4 \xi r^2 \frac {dr}{dt}[/tex]

I don't quite understand why the chain rule is being used in the first place, because [itex]4 \xi r^2[/itex] doesn't need it, and also I don't understand where [itex]\frac{dr}{dt}[/itex] comes from in the last part. I am quite confused by this basic stuff [b(] so thanks for any help.
[/tex]

Here's the fixed text for you.
 
  • #3
Originally posted by moooocow
Ok, In my calc book I am having problem with the first example they give in a chapter on Differentiation.

It gives the equation of [itex]V = \frac{4}{3} \pi r^3[/itex]

[tex]\frac{dV}{dt} = \frac {dr}{dt} \frac{dr}{dt} = 4 \pi r^2 \frac {dr}{dt}[/tex]

I don't quite understand why the chain rule is being used in the first place, because [itex]4 \pi r^2[/itex] doesn't need it, and also I don't understand where [itex]\frac{dr}{dt}[/itex]comes from in the last part. I am quite confused by this basic stuff [b(] so thanks for any help.

Your calc book seems to be implying that [itex]r[/itex] is a function of [itex]t[/itex]. So when you differentiate with respect to [itex]t[/itex] you have no choice but to use the Chain Rule.

Now clearly using the power rule we have:

[tex]
\frac{dV}{dr}=4\pi r^2
[/tex]

But we need [itex]\frac{dV}{dt}[/itex], so we invoke the chain rule:

[tex]
\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}=4\pi r^2\cdot\frac{dr}{dt}
[/tex]

We know what [itex]\frac{dV}{dr}[/itex] is so we can substitute it into the equation, but we don't know what [itex]\frac{dr}{dt}[/itex] is so it remains.
 
  • #4
the chain rule was just a mathematical trick to cancel out the "dr"'s when using 2 fractions or more.

but mathematical tricks work out great.
 
  • #5
Thanks alot, I am still having a problem understanding why exactly the chain rule is used. on the left side all you had to do was take the derivative of V and the derivative of t. I understand that [itex]\frac{dr}{dt}[/itex] is what I am trying to find, but I don't understand the reasoning behind why the chain rule is used to turn the right hand side into a derivative of time? I am a bit confused.
 
  • #6
Originally posted by moooocow
Thanks alot, I am still having a problem understanding why exactly the chain rule is used. on the left side all you had to do was take the derivative of V and the derivative of t. I understand that [itex]\frac{dr}{dt}[/itex] is what I am trying to find, but I don't understand the reasoning behind why the chain rule is used to turn the right hand side into a derivative of time? I am a bit confused.

On the left side, you aren't taking the derivative of V and the derivative of t. You're taking the derivative of V with respect to t. Basically, [itex]\frac{dV}{dt}[/itex] is a function that tells you how V changes when you change t.

On the right side, you also take the derivative with respect to t. If you were taking the derivative with respect to r, it would be easy because you can just use the power rule. But you can't, because you need the derivative with respect to t.

That's were the chain rule comes in. It let's you find the derivative of [itex]\frac{4}{3}\pi r^3[/itex] by first finding the derivative with respect to r, then multiplying that by the derivative of r with respect to t.
 
Last edited:

1. What is basic differentiation?

Basic differentiation is a mathematical concept used to find the rate of change of a function at a specific point. It involves calculating the derivative of a function, which represents the slope of the tangent line at that point.

2. Why is differentiation important?

Differentiation is important because it allows us to analyze the behavior of a function and make predictions about its values and trends. It is also essential in many fields such as physics, engineering, and economics.

3. What is the difference between basic and advanced differentiation?

Basic differentiation deals with simple functions, such as polynomials, and uses basic rules and formulas to find the derivative. Advanced differentiation involves more complex functions and requires the use of advanced techniques, such as the chain rule and product rule.

4. How is differentiation used in real life?

Differentiation has many practical applications in everyday life. For example, it is used in physics to calculate the velocity and acceleration of objects, in economics to find the marginal cost and revenue of a product, and in medicine to model the growth of tumors.

5. What are some common mistakes made in basic differentiation?

Some common mistakes in basic differentiation include forgetting to apply the chain rule or product rule, mixing up the power rule and quotient rule, and making errors in algebraic simplification. It is important to double-check your work and practice regularly to avoid these mistakes.

Similar threads

  • Calculus
Replies
16
Views
349
Replies
4
Views
187
Replies
2
Views
695
Replies
2
Views
3K
Replies
3
Views
1K
  • Calculus
Replies
2
Views
2K
Replies
2
Views
178
  • Calculus
Replies
5
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
449
Replies
22
Views
2K
Back
Top