Let y = (7/x^3)(√196x^6 +364x^2 + (169/x^2))
Find the derivative of y.
Hint: Simplify y before computing the derivative.
Answer: (-364/x^5)

- anonymous

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- anonymous

does anyone know how to simplify the equation first before taking it's derivative?

- anonymous

can you rewrite the equation using the equation tool on this site. It is tough to see in this form.

- anonymous

yep

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## More answers

- anonymous

\[y = \frac{ 7 }{ x^3 }\sqrt{196x^6 + 364x^2 + \frac{ 169 }{ x^2 }}\]

- anonymous

Answer is \[\frac{ dy }{ dx } = \frac{ -364 }{ x^5 }\]

- anonymous

I don't know how to simplify the radical. I would only know how to differentiate right away and it'll get messy

- anonymous

ok thanks anyway

- anonymous

you would use the product rule right?

- anonymous

ya.

- anonymous

I'm doing it too. it's getting pretty rough

- anonymous

\[y' = -21x ^{-4} (196x^6 + 364x^2 + 169x^{-2})^{1/2} + 7x ^{-3}(\frac{ 1 }{ 2 } (196x^6 + 364 + 169x ^{-2})^{-1/2}(1176x^5 +728x -338x ^{-3})^{-1/2}\]

- anonymous

whoops lol

- anonymous

simplifying is the hardest part

- zepdrix

Hmm there is something going on inside this square root, I'm trying to figure it out.
Not quite there yet, but I think it's not long off.
196 = 14^2
169=13^2
Hmmmm

- zepdrix

OH! yes found it! :)

- zepdrix

364 = 2 (13*14)

- zepdrix

(14x^4 + 13)^2
I think it'll simplify like that.

- anonymous

nice

- anonymous

so how would this look like

- anonymous

after simplifying?

- zepdrix

|dw:1348968835510:dw|
Hmm, I think it simplifies like this, looks much easier to deal with :)
I got lazy on the multiplying there hehe

- anonymous

thanks

- anonymous

I got the same thing. great job finding the factorization pattern @zepdrix

- anonymous

how it go from the original to this though

- anonymous

Ok I'll type this one out

- anonymous

I'm going to leave off the radical for now ok?

- anonymous

k

- anonymous

\[196x ^{6}+363x ^{2}+\frac{ 169 }{ x ^{2} }\]Find a common denominator and add\[\frac{ 196x ^{8}+364x ^{4}+169 }{ x ^{2} }\]rewrite the constants\[\frac{ 14*14x ^{8}+2*14*13x ^{4}+13*13 }{ x ^{2} }\]You still with me so far

- anonymous

notice that the numerator looks just like a polynomial

- anonymous

yep

- anonymous

\[\frac{ (14x ^{4}+13)(14x^{4}+13) }{ x ^{2}}\]

- anonymous

oh alright

- anonymous

\[\frac{ \sqrt{(14x^{4}+13)^{2}} }{ \sqrt{x^{2}} }\]

- anonymous

You probably see it now

- anonymous

thanks!

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