Question

if f(x) = 4/(x-2), find f'(x)

Answer 1

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?

Answer 2

yup

Answer 3

too bad

Answer 4

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)

Answer 5

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

Answer 6

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

Answer 7

yes.

Answer 8

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?

Answer 9

ok

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

-4h

Answer 17

exactly!

Answer 18

now recall there is an \(h\) in the denominator
cancel it

Answer 19

what is left?

Answer 20

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

Answer 21

bingo

Answer 22

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

Answer 23

-4/(x-2)^2

Answer 24

as promised (see above)

Answer 25

thank u so much! :D

Answer 26

YW!