## amy0799 one year ago if f(x) = 4/(x-2), find f'(x)

• This Question is Open
1. anonymous

i am guessing that this is the very beginning of calc and you have to do with by hand, not using any shortcuts am i right?

2. amy0799

yup

3. anonymous

4. anonymous

next week you will say $f'(x)=-\frac{4}{(x-2)^2}$ in your head, but i guess we can take the steps (long and arduous steps)

5. anonymous

we will have to go nice an slow first write the definition, then use some algebra ok a lot of algebra it is 99% algebra

6. anonymous

did you get to $\huge \lim_{h\to 0}\frac{\frac{4}{(x+h-2)}-\frac{4}{x-2}}{h}$

7. amy0799

yes.

8. anonymous

ok now in order to spare some agony lets forget about the h in the denominator, and forget about the limit we will deal with those last, ok?

9. amy0799

ok

10. anonymous

so the algebra we need is to (carefully) do this subraction $\frac{4}{x+h-2}-\frac{4}{x-2}$

11. amy0799

(4(x-2)-4(x+h-2))/((x-2)(x+h-2) is that right?

12. anonymous

can you do this? leave the denominator in factored form i.e don't multiply out

13. anonymous

yes that is correct now carefully multiply out in the numerator and combine like terms

14. amy0799

(4x-4x-8+8-4h)/((x-2)(x+h-2))

15. anonymous

ok that is multiplying out what is left in the numerator?

16. amy0799

-4h

17. anonymous

exactly!

18. anonymous

now recall there is an $$h$$ in the denominator cancel it

19. anonymous

what is left?

20. amy0799

-4/((x-2)(x+h-2))

21. anonymous

bingo

22. anonymous

now 'take the limit as h goes to zero" which is a fancy way of saying erase that h what is left?

23. amy0799

-4/(x-2)^2

24. anonymous

as promised (see above)

25. amy0799

thank u so much! :D

26. anonymous

YW!