Quotient Rule for Differentiating Functions | Homework Exercise 9

[RyanSchw]

Homework Statement

In exercises 7-12, use the Quotient Rule to differentiate the function.

9)

<br /> h(x) = \frac {\sqrt[3]{x}}{x^3+1}<br />

Homework Equations

Quotient Rule

The Attempt at a Solution

I'm trying to figure out basic calculus over the summer in preparation for class this fall, and am more or less working on my own through the odd numbered problems in the textbook. I don't think this problem is too difficult, in terms of the calculus involved, I think it's an algebra error on my part.

Apply the Quotient Rule:
<br /> \frac{(x^3+1)\frac{1}{3}x^\frac{-2}{3} - x^\frac{1}{3} (2x^2)}{(x^3+1)^2}<br />

Combine
<br /> \frac{\frac{x^3+1}{3x^2/3}-2x^7/3}{(x^3+1)^2}<br />

GCD
Since I can't seem to get this one normally..
Just omit the (x^3+1)^2 off to the right, I can't for the life of me figure out where it's coming from.

<br /> \frac{(x^{3}+1)+2x^\frac{7}{3}(3x^\frac{-2}{3})}{3x^\frac{2}{3}<br />
_____________________________
<br /> (x^3+1)^2<br />


Simplify
<br /> \frac{7x^3+1}{3x^\frac{2}{3}(x^3+1)^2}<br />

The actual answer according to both the book and my calculator
<br /> \frac{-(8x^3-1)}{3x^\frac{2}{3}(x^3+1)^2}<br />

I'm not entirely sure that I even am working in the correct direction, Any help would be greatly appreciated. Also, forgive me if I have to edit this a thousand times in order to make latex appear legible, for some reason I cannot see the image in the preview post section.

 

[VietDao29]

Apply the Quotient Rule:
<br /> \frac{(x^3+1)\frac{1}{3}x^\frac{-2}{3} - x^\frac{1}{3} (2x^2)}{(x^3+1)^2}<br />

This line is wrong, it should have read:
\frac{\frac{1}{3}(x ^ 3 + 1) x ^ {-\frac{2}{3}} - x ^ {\frac{1}{3}} (\fbox{3}x ^ 2)}{(x ^ 3 + 1) ^ 2}

(x³ + 1)' = 3x². Remember? :)

I'm not entirely sure that I even am working in the correct direction, Any help would be greatly appreciated. Also, forgive me if I have to edit this a thousand times in order to make latex appear legible, for some reason I cannot see the image in the preview post section.

Yes, the preview problem for LaTeX hasn't been fixed yet. It started with some upgrade some times ago.
But, you can still edit your post after sending it. It's a little bit inconvenient, but a way to go. :)