Question

Find the derivative of f(x) = -6 divided by x
at x = 12.

Answer 1

Another way to look at:
\[-6/x\]
is to think of it as
\[-6x^{-1}\]
That should make the derivative a bit simpler.

Answer 2

Well, I am using the difference quotient, so when doing the part involving finding f(x+h), i'm not sure how to do that with the negative exponent. What i'm really having trouble with is after setting the equation up using difference quotient:
\[\frac{\frac{ 6 }{ x + h } - (-\frac{ 6 }{ x }) }{ h }\]
I don't know what do after this.

Answer 3

\[\frac{ \frac{ -6 }{ x + h } - \frac{ -6 }{ x }}{ h }\] Careful with your signs

Answer 4

Try finding a common denominator. For instance multiple the left side by
\[\frac{x}{x}\]
and the right side by
\[\frac{x+h}{x+h}\]
that will allow you to add them and you can begin to simplify.

Answer 5

Right, sorry about that, thought I had written the negative in the first fraction :)

Answer 6

OK so,
\[\frac{ \frac{ -6x ^{2} - 6x - 6h }{ x ^{2}+hx } } { h }?\]

Answer 7

Wait a second, I messed up, it would be:
\[\frac{ \frac{ -6x + 6x + 6h }{ x ^{2}+hx } }{ h }\]
and then i'd cancel out the -6x + 6x to only have 6h on the very top top of the equation yes?

Answer 8

Right, and then you would be able to divide by h in each of the terms. What do you have left?

Answer 9

What do you mean by divide by h?
I was wondering if I could multiply by 1/h.

Answer 10

Right, same thing in this case.

Answer 11

Ok so would that cancel out the h, in 6h? and i'd be left with
6/x^2 + hx?

Answer 12

Wait no I was cross-multiplying. Okay wait, so I would multiply the entire denominator by h yes? And get
6h / hx^2 + h^2x????

Answer 13

OH WAIT I GOT IT

Answer 14

\[\frac{ 6h }{h(x^2 +hx) }\]
cancel out the h's,
and i'm left with
\[\frac{ 6 }{ x ^{2} +hx}\]

Answer 15

Ok and then, I substitute 0 into h, get 6/x^2,
replace x with 12 and my simplified answer is 1/24.
Would this be correct?

Answer 16

(1/24) is correct

Answer 17

Yes, thank you very much!

Answer 18

Sorry, I couldn't help you to the end but excellent work on your part!