Littlebird
  • Littlebird
Difference Quotient
Mathematics
katieb
  • katieb
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Littlebird
  • Littlebird
f(x) = 1/x^4
Littlebird
  • Littlebird
I've managed to get to (-4x^3-6hx^2-4h^2x-h^3)/(x^4*(x+h)^4)
triciaal
  • 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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triciaal
  • triciaal
|dw:1443157711272:dw|
Littlebird
  • Littlebird
(f(x+h)-f(x))/h
gibbs
  • gibbs
F
misty1212
  • misty1212
are you trying to take the derivative by hand ?
Littlebird
  • Littlebird
Yes
Littlebird
  • Littlebird
I know the shortcut
misty1212
  • misty1212
then you need to compute \[\lim_{h\to 0}\frac{1}{(x+h)^4}-\frac{1}{x^4}\] right?
gibbs
  • gibbs
1 Factor out the common term f f(x+h−x)/h 2 Gather like terms f((x−x)+h)/h 3 Simplify (x−x)+h to h fh/h 4. Cancel h f
ytrewqmiswi
  • ytrewqmiswi
u mean the shortcut to find the nth derivative :)
misty1212
  • misty1212
someone needs to stop @gibbs who clearly does not know what he is talking about
Littlebird
  • Littlebird
yes misty, you have what I wrote, but I also divided everything with h
Jhannybean
  • Jhannybean
Difference Quotient: \(\lim_{h\rightarrow 0} \dfrac{f(x+h) - f(x)}{h}\)
ytrewqmiswi
  • ytrewqmiswi
lml he used this XD - http://prntscr.com/8kb4j9
Jhannybean
  • Jhannybean
-_____-
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
  • 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
  • misty1212
it is now 100% algebra \[\frac{x^4-(x+4)^4}{x^4(x+h)^4}\]
misty1212
  • misty1212
you have no choice here but to expand in the numerator sorry
Littlebird
  • 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
  • Jhannybean
Ugh binomial expansion.
misty1212
  • misty1212
no choice
Jhannybean
  • Jhannybean
NO choice. you are doomed.
Littlebird
  • Littlebird
(-4x^3-6hx^2-4h^2x-h^3)/(x^4*(x+h)^4)
misty1212
  • misty1212
ooh k
Littlebird
  • Littlebird
am I normally supposed to have a limit when solving these problems?
misty1212
  • 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
  • misty1212
yes, if you are going to do it by hand
Jhannybean
  • Jhannybean
\[\dfrac{x^4-x^4-16x^3 - 96x^2-256x -256}{x^4(x+4)^4}\]
Jhannybean
  • 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
  • Jhannybean
And then you simplify the denominator there at the bottom
Littlebird
  • Littlebird
I already had it expanded. the only thing i was missing was the limit
Littlebird
  • Littlebird
you dont have to do all of this ;)
Littlebird
  • Littlebird
but thankyou
Jhannybean
  • 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
  • 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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