## anonymous 5 years ago using the definition of a derivative find f(x) =-2/sqrt{x}

1. anonymous

$f(x)= -2/ \sqrt{x}$

2. anonymous

this will be a pain to write out but not hard to compute. you must use the definition yes?

3. anonymous

i know that part i am having problems with rationalizing the denominator

4. anonymous

you need to compute $lim_{h->0}\frac{\frac{-2}{\sqrt{x+h}}+\frac{2}{\sqrt{x}}}{h}$

5. watchmath

Good opportunity to practice your LaTeX satellite! :)

6. anonymous

7. anonymous

i am learning. eventually i will be fluent in it

8. anonymous

i know that part the step after that

9. anonymous

ok lets give me a break and at least factor out the -2, since this has nothing to do with the limit ok?

10. anonymous

just factor out by 2 thats all you had t say lol thanks so much

11. anonymous

i knew that i was just doubting myself

12. anonymous

i knew that i was just doubting myself

13. anonymous

so we will just write $lim_{h->0}\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}$

14. anonymous

all divided by h of course. we worry about that last.

15. anonymous

so now we rationalize

16. anonymous

$\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}=\frac{\sqrt{x}-\sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}$

17. anonymous

yes multiply numerator and denominator by the conjugate of th enumerator.

18. anonymous

which of course is $\sqrt{x}+\sqrt{x+h}$

19. anonymous

when you do that the numerator will just be $x-x+h=h$

20. anonymous

and the denominator will be the product $\sqrt{x} \sqrt{x+h}) (\sqrt{x}+\sqrt{x+h})$

21. anonymous

sorry am not repsonding i was working it out thanks i got it

22. anonymous

thats ok i was typing. did you get the numerator correctly? it is just h

23. anonymous

yes the x's cancel

24. anonymous

the denominator is that ugly thing i wrote last. so you have in total $\frac{h}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x}+\sqrt{x+h}}$

25. anonymous

yes they 'cancel' meaning they add to zero. and of course dividing by h means the h goes in the denominator

26. anonymous

the h cancels

27. anonymous

so that last ugly thing i wrote is what you get when you rationalize the numerator. yes now the h cancels.

28. anonymous

leaving $\frac{1}{(\sqrt{x}\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})}$

29. anonymous

now you can replace h by 0 since you will not be dividing by 0.

30. anonymous

to get... $\frac{1}{(\sqrt{x}\sqrt{x})(\sqrt{x}+\sqrt{x})}$

31. anonymous

yes

32. anonymous

$=\frac{1}{2x\sqrt{x}}$

33. anonymous

oh damn i made a mistake early on the numerator was $x-(x+h)=-h$ not $x-x+h$

34. anonymous

a very bush league mistake but easily rectified. just replace the 1 in the numerator by -1

35. anonymous

sorry!

36. anonymous

so the correct answer is $\frac{-1}{2x\sqrt{x}}$

37. anonymous

then don't forget to multiply by the -2 at the end because we factored it out.

38. anonymous

so the "final answer" as we say is $\frac{1}{x\sqrt{x}}$ sorry it took a while

39. anonymous

all steps clear?

40. anonymous

sorry my computer is freezing idk if its thes site or what

41. anonymous

sometime the site is funky. earlier today it certainly was.

42. anonymous

if you have a question about any step post and i will respond

43. anonymous

i understand thank you sooo much!