Question

derivative


Find the derivative of f(x)= -9/x at x=-8

Ive gotten this far:

f(x)=-9/x
f(x+h)= -9/(x+h)

then plugging that into the derivative formula thing:

(-9/(x+h)) - (-9/x) all over h

this is where i get stuck, what do i do next

Answer 1

you have a couple of choices
one is to know that the derivative of \(\frac{1}{x}\) is \(-\frac{1}{x^2}\) so the derivative of \(\frac{-9}{x}\) is \(\frac{9}{x^2}\)

Answer 2

how does that apply to the formula thingie?

Answer 3

the other is to compute
\[\lim_{x\to -8}\frac{f(x)-f(8)}{x+8}\]

Answer 4

could i use direct substitution to compute that?

Answer 5

but we can do it your way if you like

Answer 6

yes after some algebra

Answer 7

but lets do you your way

Answer 8

ok thanks man

Answer 9

\[\frac{-9}{x+h}+\frac{9}{x}\] add them up

Answer 10

would it become

\[\frac{ -9+9(x+h) }{ x^2+h }\]

Answer 11

oh no

Answer 12

leave the denominator in factored form as \(x(x+h)\) which is not \(x^2+h\) it is \(x^2+xh\) but don't multiply out, just leave it

Answer 13

also the numerator is wrong , it should be
\[-9x+9(x+h)\]

Answer 14

yea the adding part is where i get confused

Answer 15

\[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\] is how you add fractions
the fact that in most stupid algebra problems you can find some other denominator is misleading

Answer 16

\[\frac{ 9h }{ x(x+h) }\]

?

Answer 17

yes

Answer 18

now divide by the \(h\) that we have been ignoring and get
\[\frac{9}{x(x+h)}\]

Answer 19

what would we do next?

Answer 20

take the limit as \(h\) goes to zero, which really means erase the \(h\)
then replace \(x\) by \(-8\) or any other number of your choosing

Answer 21

so we just plug in -8 for x and 0 for h?

Answer 22

first get rid of the \(h\) and you get
\[f'(x)=\frac{9}{x^2}\] as promised way above
then you can compute
\[f'(-8)=\frac{9}{(-8)^2}\] or \[f'(whatever)\] because now you have an explicit expression for \(f'\)

Answer 23

so we end up with 9/64 as the derivative?

Answer 24

yes

Answer 25

thanks SO much!!

Answer 26

if this seems hard or annoying, don't worry in a week you will be able to do this in your head
or maybe two weeks
yw