Question

Find the derivative by the limit process.
9/x^2
I am writing what I have done so far but I do have the answer already, just not the work yet.
Answer is -18/x^3

Answer 1

Okay first I put it into the formula,\[\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0} (\frac{9}{x+\Delta x}-\frac{9}{x^2})*\frac{1}{\Delta x}\]Then I got the denominators the same in the parenthesis so I could simplify it and factor out the Δx,\[\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0} (\frac{9(x^2)}{(x+\Delta x)(x^2)}-\frac{9(x+\Delta x)}{(x^2)(x+\Delta x)})*\frac{1}{\Delta x}\]=\[\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0} (\frac{9(x^2)-9(x+\Delta x)}{(x+\Delta x)(x^2)})*\frac{1}{\Delta x}\]This is where I got stuck. I tried to multiple the top out but it didn't help. Generally at this point I would cancel a Δx somewhere but I can't on this one.. Any ideas?

Answer 2

your steps look pretty good
what you forgot what that the (x+dx) should be squared as well

Answer 3

Typo, it is squared in my actual notes.

Answer 4

\[f(x) = \frac{9}{x^{2}}\]
\[f(x+\Delta x) = \frac{9}{(x+\Delta x)^{2}}\]
oh :|

Answer 5

sorry but thanks for catching it :)

Answer 6

in that case, by multiplying the top and simplifying the "x^2" terms should cancel and then you can factor out a dx

Answer 7

\[\rightarrow 9x^{2}-9(x^{2}+2x \Delta x +\Delta x^{2}) = -18x \Delta x-9 \Delta x^{2} = \Delta x(-18x-9 \Delta x)\]

Answer 8

I made a mistake in my notes but yes I got the same thing and the Δx in front cancels with the 1/Δx. I'm working on cleaning the rest up now, give me minute...

Answer 9

correct..now it should work out nicely

Answer 10

Okay I got it.\[\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0} \frac{-18x-9\Delta x}{x^2(x+\Delta x)}\rightarrow \Delta x = 0\rightarrow \lim_{\Delta x \rightarrow 0}\frac{-18x}{x(x^3)}\]I skipped some steps in that last part but I did multiply everything out and then dropped all the Δx's. Cancel and it leaves me with the correct answer of \[\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0} -\frac{18}{x^3}\]

Answer 11

:)

Answer 12

pretty soon you won't have to do that process anymore...they will teach you an easier rule to find derivatives