## babz83 4 years ago find dx/dy for the following equations 1.) y = square root of 3 (x+1)/x-1)

1. babz83

$\sqrt[3]{X+1/x-1}$

2. babz83

2) y = cos(lnX)

3. kushashwa23

ok

4. kushashwa23

let me have a look to this closely

5. babz83

6. Owlfred

7. amistre64

cbrt((x+1)/(x-1)) right?

8. amistre64

change the radical to its proper exponent form [ (x+1)/(x-1) ]^(1/3) and work thru

9. amistre64

this aint gonna be nice with my formatting off

10. amistre64

lets start with the ^1/3; power rule says bring it down and -1

11. babz83

@owlfred...nope remember she said she got an A in calc.... i would not be posting these questions

12. amistre64

(1/3) [ (x+1)/(x-1) ]^(-2/3) and out pops the innards for the chain rule

13. amistre64

Dx(x+1)/(x-1) = (x-1)(1) - (x+1)(-1) // (x-1)^2 by quotient rule

14. amistre64

simplify to: 2x/(x-1)^2 right?

15. amistre64

put it all together to go: 2x/3(x-1)^2) [ (x-1)/(x+1) ]^(2/3)

16. babz83

am following u amistre

17. babz83

let me just digest it!

18. amistre64

$\frac{2x}{3(x-1)^2}\left(\frac{x-1}{x+1}\right)^{2/3}$right?

19. amistre64

ill type it in the latex stuff fo try to make it easier to follow

20. babz83

nope

21. babz83

$\sqrt[3]{x+1/x-1}$

22. babz83

ah i get it thats after removing the cube root

23. amistre64

$\sqrt[3]{\frac{x+1}{x-1}}$ $Dx\left(\frac{x+1}{x-1}\right)^{1/3}$ $\frac{1}{3}\left(\frac{x+1}{x-1}\right)^{-2/3}Dx(\frac{x+1}{x-1})$ so far

24. amistre64

-2/3 just flips it all over and goes positive again right?

25. babz83

right

26. amistre64

$\frac{1}{3}\left(\frac{x-1}{x+1}\right)^{2/3}Dx(\frac{x+1}{x-1})$ $... Dx(\frac{x+1}{x-1})=\frac{bt'-b't}{b^2}$ $... \frac{(x-1)(1)-(x+1)(-1)}{(x-1)^2}$ $... \frac{(x-1)+(x+1)}{(x-1)^2}$ $... \frac{x-1+x+1}{(x-1)^2}$ $... \frac{2x}{(x-1)^2}$

27. amistre64

$\frac{1}{3}\left(\frac{x-1}{x+1}\right)^{2/3}\frac{2x}{(x-1)^2}$ are all the parts, just simplify as wanted

28. amistre64

i see a spot i missed; (x-1) has a derivative if 1, not -1

29. babz83

thnks a lot

30. babz83

i see i have a lot of serious reading to do

31. amistre64

$... \frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}$ $... \frac{x-1-x-1}{(x-1)^2}$ $... \frac{-2}{(x-1)^2}$ $\frac{1}{3}\left(\frac{x-1}{x+1}\right)^{2/3}\frac{-2}{(x-1)^2}$ thats proper, or should be better at least :)