Question

Find the derivative of f(x) = 6 divided by x at x = -2.

Answer 1

Hey Josh, welcome to OpenStudy!

Recall that we have a handy exponent rule to help us out on problems like this,
\(\large\rm \dfrac{1}{x}=x^{-1}\)
From there we can apply our `power rule`.

Have you learned about power rule yet?
Or do we need to use the limit definition of the derivative at this point?

Answer 2

no i havent

Answer 3

I want to say the answer is -3/2 but im not sure if its correct

Answer 4

Ok so we'll have some tricky algebra to take care of then.\[\large\rm f(\color{orangered}{x})=\frac{6}{\color{orangered}{x}}\]
If we evaluate this function at x+h, instead of x, it looks like this\[\large\rm f(\color{orangered}{x+h})=\frac{6}{\color{orangered}{x+h}}\]

Answer 5

\[\large\rm \lim_{h\to0}\frac{f(x+h)-f(x)}{h}\quad=\quad \lim_{h\to0}\frac{\frac{6}{x+h}-\frac{6}{x}}{h}\]

Answer 6

so then its not negative its just 3/2

Answer 7

I dunno, you're jumping straight to the end lol :)

Answer 8

so what numbers do i input

Answer 9

\[\large\rm f'(-2)\quad=\quad \lim_{h\to0}\frac{\frac{6}{-2+h}-\frac{6}{-2}}{h}\]I guess -2 is our input.
You'll want to do something with the numerator,
maybe common denominator, ya?

Answer 10

i dont know how to do that

Answer 11

The derivative of 1/x is -1/x squared
Multiply by the constant 6 and you havevthe general derivative. Then replace the x by -2

Answer 12

ohhhh okay! give me a second

Answer 13

so (-1/-2) * 6

Answer 14

Yeah but the -2 is squared

Answer 15

you cant have exponents on the bottom though?

Answer 16

Yes you can
If you have exponent -2 for example on top, you can transfer it to the bottom but it becomes 2. The sign of the exponent changes but you can have them both on top or bottom

Answer 17

so the answer is 6!

Answer 18

The answer is 6 *(-1/(-2)squared)

Answer 19

So -6/4

Answer 20

So -3/2

Answer 21

awesome thank you!

Answer 22

So your first answer was correct

Answer 23

You're welcome :)