## anonymous 5 years ago trying to find the second derivative in this question......3x-3 over 2x+4. the Vertical Asy is -2 and the Horizontal Asy is 3/2...the xint is (1,0) and the yint is (0,-3/4) please give me step buy step

1. anonymous

can you find the first derivative? because that is the only hard part

2. anonymous

quotient rule give $\frac{d}{dx}\frac{3x-3}{2x+4}=\frac{(2x+4)\times 3-(3x+3)\times 2}{(2x+4)^2}$

3. anonymous

yes i have the first one all ready

4. anonymous

so you end up with $\frac{9}{(2x+4)^2}=9(2x+4)^{-2}$ now second should be easy

5. anonymous

$-18(2x+4)^{-3}\times 2=-36(2x+4)^{-3}=-\frac{36}{(2x+4)^3}$

6. anonymous

you can cancel a little because $(2x+4)^3=(2(x+2))^3=8(x+2)^3$

7. anonymous

im so confused

8. anonymous

ok which step? did you get the first derivative?

9. anonymous

i got (2x+4)-2(3x-3) over (2x+4)^2

10. anonymous

then you have to do the algebra in the numerator

11. anonymous

should be $(2x+4)3-2(3x-3)$

12. anonymous

in front of the 2x+4 there is suppose to be a 3

13. anonymous

yes it is the denominator times the derivative of the numerator as the first term

14. anonymous

minus the numerator times the derivative of the denominator

15. anonymous

if you multiply out the x terms add up to zero and you get 9

16. anonymous

so 18 over (2x+4)^2 ?

17. anonymous

yes

18. anonymous

$(2x+4)3-2(3x-3)$ $6x+12-6x+6=18$ yes

19. anonymous

my equation is 3(2x+4)-2(3x-3)

20. anonymous

yes multiply out and you will get 18

21. anonymous

ok then what do i do to get the second answer

22. anonymous

you have $\frac{18}{(2x+4)^2}$ so rather than using the quotient rule again, rewrite in exponential form as $18\times (2x+4)^{-2}$ and use the power rule (and the chain rule)

23. anonymous

ok so multiple it out?

24. anonymous

power rule right? $\frac{d}{dx}x^n=nx^{n-1}$

25. anonymous

26. anonymous

$\frac{d}{dx}18(2x+4)^{-2}=-2\times 18(2x+4)^{-2-1}\times 2$ $=-72(2x+4)^{-3}=-\frac{72}{(2x+4)^3}$ but you can simplify this

27. anonymous

because $(2x+4)^3=(2(x+2))^3=8(x+2)^3$ and $\frac{72}{8}=9$

28. anonymous

so your "final answer" will be $-\frac{9}{(x+2)^3}$

29. anonymous

thank you so much

30. anonymous

yw