Quotient Rule problem: Differentiate f(x) = (x^2 + 4x + 6) /x​^{1/2}

[ricecrispie]

Hi been at this question for a while now, will really appreciate any help!

Differentiate:

f(x) = (x² + 4x + 6) /x^​1/2

 

[tkhunny]

1) Please state the "Quotient Rule".
2) Please demonstrate how you have implemented it.

 

[Dr.Peterson]

Hi been at this question for a while now, will really appreciate any help!

Differentiate:

f(x) = (x² + 4x + 6) /x^​1/2

Presumably you have tried using the quotient rule; we'll need to see your work in order to be sure what help you need.

I would do it differently. Note that you can distribute the division, dividing each term:

x²/x^​1/2 + 4x/x^​1/2 + 6/x^​1/2

Simplify, then differentiate. Again, please show your work so we can know where, if at all, you are going wrong. The difficulty doing it this way may be to convince yourself that your answer agrees with the book's!

(By the way, thanks for writing the function clearly, parentheses and all!)

 

[ricecrispie]

Hi, I've just solved this problem thanks!

 

[ricecrispie]

Hi yes I did that and it worked out, thanks for the help!

 

[sinx]

Presumably you have tried using the quotient rule; we'll need to see your work in order to be sure what help you need.

I would do it differently. Note that you can distribute the division, dividing each term:

x²/x^​1/2 + 4x/x^​1/2 + 6/x^​1/2

Simplify, then differentiate. Again, please show your work so we can know where, if at all, you are going wrong. The difficulty doing it this way may be to convince yourself that your answer agrees with the book's!

(By the way, thanks for writing the function clearly, parentheses and all!)

good show, Dr Peterson.
It occured to me that from here you could rewrite each term as a product, [and differentiate using the product rule].
i.e. x²(x^​-1/2 )+ 4x(x^​-1/2 ) + 6(x^​-1/2 )
obviously the first two terms can be simplified before differentiating.

I don't offer this as an answer to the original question.
however, it should be insightful to check the product rule soln with the soln by quotient rule.

 

[Dr.Peterson]

good show, Dr Peterson.
It occured to me that from here you could rewrite each term as a product, [and differentiate using the product rule].
i.e. x²(x^​-1/2 )+ 4x(x^​-1/2 ) + 6(x^​-1/2 )
obviously the first two terms can be simplified before differentiating.

I don't offer this as an answer to the original question.
however, it should be insightful to check the product rule soln with the soln by quotient rule.

Of course, you don't even need the product rule, if you simplify each term first.

I tend to avoid the quotient rule in practice, because I'm never sure if I've got the order right if I haven't done much lately. So I do what you have done with each term here, rewriting a quotient as a product when possible (or else I derive the quotient rule that way to remind me which way it works). And when I can avoid the product rule too, all the better.

Of course, someone learning the quotient rule, as in this case, should use it for the sake of practice, and then use another method as a check when appropriate. You're entirely right there. Solving a problem two different ways is a great habit to develop when learning.