Application of derivative rules in physics

[physics_who]

Homework Statement

Hi everyone, I'm currently working on year 12 maths and am able to answer questions in the maths book for the various rules of differentiation (chain, product, quotient) and can determine which questions should be answered using which rules.

But in the maths book, the questions are presented in the form of equations and there are no word questions in the book I'm using.

The issue I'm having is trying to come up with examples in physics where the quotient rule would be used - like coming up with a worked solution question that requires using the quotient rule in order to solve it (that would be accessible at the year 12 level). Would anyone be able to provide such an example?

While I can solve the maths questions, I don't feel like I can really understand it until I can see it in action, if that makes sense.

Homework Equations

The quotient rule:
dy/dx = [v(du/dx) - u(dv/dx)] / v^2

The Attempt at a Solution

I've done some extensive web searches and have found some good examples for the chain rule and the product rule, but haven't been able to find anything for the quotient rule.

I tried to create my own question using basic kinematics, but haven't had much success.

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/chainap.html <- this is a good example of what I'm looking for with regard to the quotient rule, which will hopefully make it clear what I'm hoping to find.

Any help would be much appreciated. Thanks!

 

[RUber]

In reality, the quotient rule and the product rule are the same...it is just a special case. So, I recommend not treating them like different rules.
If you need to verify, try looking at ## f(x) = \frac{u(x)}{v(x)} ## as the product ##f(x) = u(x) * (v(x))^{-1}##.
This will come up any time it is easier for you to look at a function as a fraction of two other functions, and is simply a matter of convenience.

 

[David Lewis]

The angle of attack (α) that corresponds to maximum lift-to-drag occurs where the derivative of the lift function divided by the drag function (with respect to α) equals zero.

 

[physics_who]

In reality, the quotient rule and the product rule are the same...it is just a special case. So, I recommend not treating them like different rules.
If you need to verify, try looking at ## f(x) = \frac{u(x)}{v(x)} ## as the product ##f(x) = u(x) * (v(x))^{-1}##.
This will come up any time it is easier for you to look at a function as a fraction of two other functions, and is simply a matter of convenience.

Thank you for this, that's very helpful. I do understand your point, but am still hoping to find a worked example for the quotient rule. :)

The angle of attack (α) that corresponds to maximum lift-to-drag occurs where the derivative of the lift function divided by the drag function (with respect to α) equals zero.

Thank you, David, this is along the lines of what I'm looking for. I'm currently looking up these formulae to try and construct my own worked example, but would certainly appreciate if anyone would be able to provide a hypothetical example.

 

[FactChecker]

Calculate the jerk of something whose mass is changing due to burning fuel (a rocket, car, etc.): A = F(t)/m(t); jerk = A'.

Calculate the rate of change of pressure in a cylinder with a moving piston.

A situation where you would need to find those derivatives are if you are trying to control acceleration or pressure and want to know how often you need to monitor the acceleration or pressure.
EDIT Correction: replaced "jerk" with "acceleration" in prior line.

 

[physics_who]

The angle of attack (α) that corresponds to maximum lift-to-drag occurs where the derivative of the lift function divided by the drag function (with respect to α) equals zero.

I was able to find this page that shows the derivation of L/Dmax, which does involve the quotient rule. Thank you!

 

[physics_who]

Calculate the jerk of something whose mass is changing due to burning fuel (a rocket, car, etc.): A = F(t)/m(t); jerk = A'.

Calculate the rate of change of pressure in a cylinder with a moving piston.

A situation where you would need to find those derivatives are if you are trying to control jerk or pressure and want to know how often you need to monitor the jerk or pressure.

Thank you for this - I will have a look at these calculations and try to work something out. :)

 

[Ray Vickson]

Homework Statement

Hi everyone, I'm currently working on year 12 maths and am able to answer questions in the maths book for the various rules of differentiation (chain, product, quotient) and can determine which questions should be answered using which rules.

But in the maths book, the questions are presented in the form of equations and there are no word questions in the book I'm using.

The issue I'm having is trying to come up with examples in physics where the quotient rule would be used - like coming up with a worked solution question that requires using the quotient rule in order to solve it (that would be accessible at the year 12 level). Would anyone be able to provide such an example?

While I can solve the maths questions, I don't feel like I can really understand it until I can see it in action, if that makes sense.

Homework Equations

The quotient rule:
dy/dx = [v(du/dx) - u(dv/dx)] / v^2

The Attempt at a Solution

I've done some extensive web searches and have found some good examples for the chain rule and the product rule, but haven't been able to find anything for the quotient rule.

I tried to create my own question using basic kinematics, but haven't had much success.

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/chainap.html <- this is a good example of what I'm looking for with regard to the quotient rule, which will hopefully make it clear what I'm hoping to find.

Any help would be much appreciated. Thanks!

Look up Planck's Law (for the distribution of power vs frequency in blackbody radiation). To find the frequency giving maximum power involves exactly the type of thing you want.

 

[FactChecker]

Just think of things in physics where you do some division (estimated_time_to_reach_target=distance/velocity; density = mass/volume; etc.) and imagine situations where the divisor changes with time. You can probably think of many examples.

 

[David Lewis]

Good points. And you also need a reason to want to differentiate the quotient of two functions. In the example I offered, the idea is to find the maximum (derivative = zero). In other cases, you might just need a general expression for the rate of change of the ratio of two functions.