anonymous one year ago Find the derivative of f(x) = 6/x at x = -2.

1. anonymous

solomon help haha

2. SolomonZelman

Ok, lets re-write the f(x) $$\large\color{black}{ \displaystyle f(x)=6(x)^{-1} }$$

3. SolomonZelman

Apply the power rule. Can you do that?

4. SolomonZelman

(YOu are to find the derivative, and then plug in x=-2 into the derivative)

5. anonymous

-3

6. SolomonZelman

7. SolomonZelman

if so, then you are not correct....

8. SolomonZelman

Did you find the $$f'(x)$$ /?

9. SolomonZelman

Oh, what I mean by the power rule is: $$\large\color{black}{ \displaystyle \frac{d }{dx}x^n=nx^{n-1} }$$ where d/dx is jst a notation for taking the derivative. ------------------------------------------------ But I guess you are doing by the first principles...

10. anonymous

'm not really sure what you mean by power rule I have a formula for difference quotient f(h-1)-f(1)/h and I ended up with ([6/h-1]-6)/h

11. anonymous

never heard of it this is precalc

12. SolomonZelman

yes, you are applying the following: $$\large\color{black}{ \displaystyle \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} }$$

13. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} }$$

14. SolomonZelman

YOu can use the power rule I posted, to at least check the work, but for now I guess we need this: $$\large\color{black}{ \displaystyle \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} }$$

15. SolomonZelman

Or, if you want to find $$f'(a)$$ direclty: $$\large\color{black}{ \displaystyle f'(a)= \lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a} }$$

16. anonymous

(6/x+h)-(6/x)/h

17. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{6}{x+h}-\dfrac{6}{x} }{h} }$$

18. SolomonZelman

that is right.

19. SolomonZelman

now, find the common denominator betwen 6/(x+h) and 6/x and subtract.

20. anonymous

so would I multiply one side by x and the other by x+h

21. SolomonZelman

yes, fraction#1 by x on top and bottom, and fraction#2 (x+h) on top and bottom

22. anonymous

I got -6

23. anonymous

pretty sure that's it thanks

24. SolomonZelman

no it is not it

25. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{6}{x+h}-\dfrac{6}{x} }{h} }$$ $$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{6x}{x(x+h)}-\dfrac{6(x+h)}{x(x+h)} }{h} }$$

26. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{6x-6(x+h)}{x(x+h)} }{h} }$$

27. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{-6h}{x(x+h)} }{h} }$$

28. anonymous

yep that's what I got just forgot about the denomenator under the -6h

29. SolomonZelman

then divide top and bottom by h.

30. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}\frac{\dfrac{-6h}{x(x+h)}\color{red}{\div h} }{h \color{red}{\div h}} }$$

31. SolomonZelman

then you get: $$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}{~} \frac{-6}{x(x+h)} }$$

32. anonymous

-6/x(x+h)

33. SolomonZelman

yes, but you are leaving out that important limit.

34. anonymous

so I just put -2 in for x

35. SolomonZelman

that limit that h=0, is an important component. So that when you simplify the expression, you then plug in h=0 (if you don;t get any undefined results for that)

36. SolomonZelman

$$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}{~} \frac{-6}{x(x+h)} =\frac{-6}{x(x+0)}=\frac{-6}{x^2} }$$

37. SolomonZelman

see what is that limit for? (it is a notation for the fact that h is 0)

38. anonymous

- 3/2

39. SolomonZelman

yes, that is right:

40. anonymous

hey, thanks for the patience I'm kind of slow

41. SolomonZelman

Yes, don't forget that limit h->0 notation. it is important.

42. SolomonZelman

So just an addition that in general: $$\large\color{black}{ \displaystyle f'(x)= \lim_{h \rightarrow 0}{~} \frac{f(x+h)-f(x)}{h} }$$ (Derivative a function f(x).) $$\large\color{black}{ \displaystyle f'(x)= \lim_{x \rightarrow a}{~} \frac{f(x)-f(a)}{x-a} }$$ (Derivative a function f(x) evaluated at x=a.)

43. SolomonZelman

good luck