Differentiation Question - Chain Rule

[BioBabe91]

Homework Statement

Differentiate y = $$\left(\frac{x+2}{\sqrt[3]{x}}\right)$$³

Homework Equations

-Chain Rule
-Quotient Rule
-Power Rule
-Product Rule?

The Attempt at a Solution

First I got rid of the fraction by taking the negative of x^3, and then used the chain rule to differentiate. However, it just got very complicated, and I can't solve it because it's way too complicated. Is there a short way to do this?

 

[rock.freak667]

First I got rid of the fraction by taking the negative of x^3, and then used the chain rule to differentiate. However, it just got very complicated, and I can't solve it because it's way too complicated. Is there a short way to do this?

What do you mean by you took the negative of x³?

Show what you did but most likely there isn't a shorter way to do it.

 

[Mark44]

You can get rid of the fraction by writing the x^(1/3) in the denominator as x^(-1/3) in the numerator.

Then, to differentiate, you'll need to use the
chain rule and then the product rule, in that order.

 

[BioBabe91]

I did what Mark44 said, then I used the chain rule and the product rule. This is how I did it:

But what do I do next? Do I have to do all the multiplication to bring everything under a single denominator? Because I don't seem to be getting the right answer that way... did I do everything right?
The answer should be $$\frac{\left(1+\sqrt{x}\right)^{2}\left(\sqrt{x}-2\left)}{x^{3}}$$

 

[Mark44]

I did what Mark44 said, then I used the chain rule and the product rule. This is how I did it:

But what do I do next? Do I have to do all the multiplication to bring everything under a single denominator? Because I don't seem to be getting the right answer that way... did I do everything right?
The answer should be $$\frac{\left(1+\sqrt{x}\right)^{2}\left(\sqrt{x}-2\left)}{x^{3}}$$

The 3rd line is the derivative you want, and you can stop there. Anything else you do after that is algebraic manipulation. I should point out in this line that you need a pair of parentheses surrounding everything after the dot, and the same is true for the two lines after that.

You have a mistake in the 4th line where you cancel 3s, and this is related in a way to the lack of parentheses around the terms following the dot.

Are you sure that the answer you show as the right one goes with this problem? Are you sure that you worked the same problem? I don't see any connection between your "correct" answer and this problem.

 

[BioBabe91]

wait... yes, sorry. i did copy the wrong one. so the answer for this one is $$\frac{2\left(x+2\right)^{2}\left(x-1\left)}{x^{2}}$$. But I'm still not getting the answer...

 

[AEM]

Homework Statement

Differentiate y = $$\left(\frac{x+2}{\sqrt[3]{x}}\right)$$³

Homework Equations

-Chain Rule
-Quotient Rule
-Power Rule
-Product Rule?

The Attempt at a Solution

First I got rid of the fraction by taking the negative of x^3, and then used the chain rule to differentiate. However, it just got very complicated, and I can't solve it because it's way too complicated. Is there a short way to do this?

Yes there is a shorter way to do this. You can write it as

$$\frac{(x+2)^3}{(\sqrt[3]{x})^3}$$

and notice that that leaves you with

$$\frac{(x+2)^3}{x}$$

which is considerably easier to differentiate.

 

[BioBabe91]

That definitely worked! Thanks :)