## iop360 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. iop360

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

2. ChmE

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

3. iop360

yep

4. iop360

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

5. iop360

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

6. ChmE

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

7. iop360

ok thanks anyway

8. iop360

you would use the product rule right?

9. ChmE

ya.

10. ChmE

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

11. iop360

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

12. iop360

whoops lol

13. iop360

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. ChmE

nice

19. iop360

so how would this look like

20. iop360

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. iop360

thanks

23. ChmE

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

24. iop360

how it go from the original to this though

25. ChmE

Ok I'll type this one out

26. ChmE

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

27. iop360

k

28. ChmE

$196x ^{6}+363x ^{2}+\frac{ 169 }{ x ^{2} }$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. ChmE

notice that the numerator looks just like a polynomial

30. iop360

yep

31. ChmE

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

32. iop360

oh alright

33. ChmE

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

34. ChmE

You probably see it now

35. iop360

thanks!