Should I Use the Quotient Rule for Derivatives of Fractions?

[Liquid7800]

Homework Statement

I would like to know...when trying to take the derivative of a function with a fraction in it ...
should I always turn it into a product and use the product rule, thereby dropping the quotient rule most of the time?
Or is the quotient rule needed more so in some cases?

Its like once you know the power rule etc. why take the derivative using the Definition or Newton Quotient, you can do it...but why would you want to?

Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[danago]

I think it depends on the question. I find that it is easier to use the quotient rule in some cases, and product rule in others. They are both really different forms of the same rule (i.e. you can easily derive one from the other), but i don't think you should just forget about one completely; practice using both.

 

[Liquid7800]

Thanks for the good advice...I plan on practicing both...and I suppose it does depend on the question (ie finding the derivative with the definition...)

Now last question (sorry I am a newbie at calculus)...

whether you use the product rule for a derivative with a fraction in the function, by changing it to a product, or the quotient rule and leave it alone...

both WILL come out to the same answer always right?


Thanks for the reply too.

 

[danago]

If you do it correctly, then yea, they will both yield the same answer.

 

[epenguin]

I agree with danago.

The convenience might depend what you want to do with it at the end - although you might not be able to see that at the beginning.

e.g. the first term in the quotient form is vu'/v² so there is a cancellation where the product form gives you u'v^-1 straight for the first term, on the other hand you might be interested in the numerator of the quotient form as a whole for some other factorisation for example. There isn't much in it

 

[Liquid7800]

Good advice everyone:

I went into my test after studying both ways and realized that I would use one way or the other depending on how the other parts of the expression surrounding a particular part were expressed...eg

If most of the terms were products...I would convert to product as well...if most of them were quotients I would convert to quotients etc...also I find the Quotient rule is good for discerning trig identities easier then converting to product rules.

 

[crd]

I find that the quotient rule is easier to work with algebraically, since you already have the lcd(which is the first step of simplifying a derivative found by the using the product on quotient.) Also, if you are interested in zeros/critical points, the quotient rule saves you steps on your algebra as well.

 

[wintsmith]

You might also look at the derivative in the form (u' - f v')/v where f=u/v. I discuss this form of the derivative, and a property of critical points, on my blog at:


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