Difference Quotient

- Littlebird

Difference Quotient

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- Littlebird

f(x) = 1/x^4

- Littlebird

I've managed to get to (-4x^3-6hx^2-4h^2x-h^3)/(x^4*(x+h)^4)

- triciaal

what do you mean by difference quotient?
difference what you get when you subtract
quotient what you get when you divide

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## More answers

- triciaal

|dw:1443157711272:dw|

- Littlebird

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

- gibbs

F

- misty1212

are you trying to take the derivative by hand ?

- Littlebird

Yes

- Littlebird

I know the shortcut

- misty1212

then you need to compute
\[\lim_{h\to 0}\frac{1}{(x+h)^4}-\frac{1}{x^4}\] right?

- gibbs

1 Factor out the common term f
f(x+hâˆ’x)/h
2 Gather like terms
f((xâˆ’x)+h)/h
3 Simplify (xâˆ’x)+h to h
fh/h
4. Cancel h
f

- ytrewqmiswi

u mean the shortcut to find the nth derivative :)

- misty1212

someone needs to stop @gibbs who clearly does not know what he is talking about

- Littlebird

yes misty, you have what I wrote, but I also divided everything with h

- Jhannybean

Difference Quotient: \(\lim_{h\rightarrow 0} \dfrac{f(x+h) - f(x)}{h}\)

- ytrewqmiswi

lml he used this XD - http://prntscr.com/8kb4j9

- Jhannybean

-_____-

- Jhannybean

So find all the hard stuff first: \(f(x+h) = \dfrac{1}{(x+h)^4}\)
put it back into the formula : \(\dfrac{\dfrac{1}{(x+h)^4} - \dfrac{1}{x^4}}{h}\)
Then start simplifying by finding a common denominator in the numerator portion.

- misty1212

ok leave out the \(h\) in the denominator for a moment and focus just on \[\frac{1}{(x+h)^4}-\frac{1}{x^4}\] we will divide by ]\(h\) at the end

- misty1212

it is now 100% algebra \[\frac{x^4-(x+4)^4}{x^4(x+h)^4}\]

- misty1212

you have no choice here but to expand in the numerator
sorry

- Littlebird

I just noticed that misty put a limit of h->0 next to the thing before
If I use that I actually get the answer

- Jhannybean

Ugh binomial expansion.

- misty1212

no choice

- Jhannybean

NO choice. you are doomed.

- Littlebird

(-4x^3-6hx^2-4h^2x-h^3)/(x^4*(x+h)^4)

- misty1212

ooh k

- Littlebird

am I normally supposed to have a limit when solving these problems?

- misty1212

looks like you cancel the \(h\) to right? so now all that is left is to replace \(h\) by \(0\) and you are done

- misty1212

yes, if you are going to do it by hand

- Jhannybean

\[\dfrac{x^4-x^4-16x^3 - 96x^2-256x -256}{x^4(x+4)^4}\]

- Jhannybean

It's just too much to constantly write the limit over and over again in latex, so i just leave it for the end... it's wrong but <_<

- Jhannybean

And then you simplify the denominator there at the bottom

- Littlebird

I already had it expanded. the only thing i was missing was the limit

- Littlebird

you dont have to do all of this ;)

- Littlebird

but thankyou

- Jhannybean

Why have you multiplied h to the numerator portion?
\(\color{#0cbb34}{\text{Originally Posted by}}\) @Littlebird
\(\color{red}{(-4x^3-6hx^2-4h^2x-h^3)}/(x^4*(x+h)^4)
\(\color{#0cbb34}{\text{End of Quote}}\)
That portion.

- Littlebird

I multiplied both the top and bottom of the fraction by x^4(x+h)^4 so that I could make the fractions on top go away

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