## anonymous 3 years ago 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)

1. anonymous

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

2. anonymous

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

3. anonymous

yep

4. anonymous

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

5. anonymous

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

6. anonymous

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

7. anonymous

ok thanks anyway

8. anonymous

you would use the product rule right?

9. anonymous

ya.

10. anonymous

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

11. anonymous

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

12. anonymous

whoops lol

13. anonymous

simplifying is the hardest part

14. 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

15. zepdrix

OH! yes found it! :)

16. zepdrix

364 = 2 (13*14)

17. zepdrix

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

18. anonymous

nice

19. anonymous

so how would this look like

20. anonymous

after simplifying?

21. 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

22. anonymous

thanks

23. anonymous

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

24. anonymous

how it go from the original to this though

25. anonymous

Ok I'll type this one out

26. anonymous

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

27. anonymous

k

28. 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

29. anonymous

notice that the numerator looks just like a polynomial

30. anonymous

yep

31. anonymous

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

32. anonymous

oh alright

33. anonymous

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

34. anonymous

You probably see it now

35. anonymous

thanks!