# Thread: Need help with these derivatives

1. ## Need help with these derivatives

, show that ; for this problem I got

, show that ; for this problem I got

, show that ; for this problem I got

, show that ; for this problem I got

, show that ; for this problem I got

I know how to get the first derivatives for all these problems, but I'm having problem getting to these alternate derivative forms. Please help. Thanks.

2. ## Re: Need help with these derivatives

Then sow us what you got please...

3. ## Re: Need help with these derivatives

Sorry. I edited the post and included the solutions that I have so far.

4. ## Re: Need help with these derivatives

Hello, rainbowbaconz!

$\displaystyle f(x) \:=\:\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^3$

$\displaystyle \text{Show that: }\:f'(x) \:=\:\frac{3x^3+3x^2-3x-3}{2x^2\sqrt{x}}$ . What a STUPID way to write the answer!

We have: .$\displaystyle f(x) \:=\:\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^3$

Then: .$\displaystyle f'(x) \;=\;3\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2\left(\tfrac{1}{2}x^{-\frac{1}{2}} - \tfrac{1}{2}x^{-\frac{3}{2}}\right)$

. . . . . $\displaystyle f'(x) \;=\; 3\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2\cdot\tfrac{1}{2}\left( \frac{1}{x^{\frac{1}{2}}} - \frac{1}{x^{\frac{3}{2}}}\right)$

. . . . . $\displaystyle f'(x) \;=\;\frac{3}{2}\left(\frac{x+1}{\sqrt{x}}\right)^ 2\left(\frac{x-1}{x^{\frac{3}{2}}}\right)$

. . . . . $\displaystyle f'(x) \;=\;\frac{3}{2}\cdot\frac{(x+1)^2}{x}\cdot\frac{x-1}{x^{\frac{3}{2}}}$

. . . . . $\displaystyle f'(x) \;=\;\frac{3(x+1)^2(x-1)}{2x^{\frac{5}{2}}}$

5. ## Re: Need help with these derivatives

Thanks for the reply Soroban. I understand the process, but how would you go about getting to the alternative form that I posted for that problem?