Question

f(x)= 1/x^7 find the derivativeusing the limiting process. Help, I understand some of it, i know the formula but i get lost, and need help solving these

Answer 1

how much you understand?

Answer 2

\[\lim_{h->0}\frac{f(a+h)-f(a)}{h}\]

Answer 3

my teacher uses c and delta x instead of a and h, i dont know why. I understand it up untill the point where u find f(c + delta x) then when im writing it out, i make errors. -_- It must be my algebra skills are iffy, since its been a long time sicne ive taken algebra, but my work never looks "right"

Answer 4

\[\lim_{h->0}\frac{\cfrac{1}{(a+h)^7} -\cfrac{1}{(a)^7}}{h}\]
\[\lim_{h->0}\frac{\cfrac{a^7-(a+h)^7}{(a+h)^7}}{h}\]
\[\lim_{h->0}\frac{a^7-(a+h)^7}{h(a+h)^7}\]
now the trick is to expand the top to get out a factor of h to cancel with

Answer 5

\[(a+h)^7 = a^7+ 7a^6h +21a^5h^2\]\[=> +35a^4h^3+ 35a^3h^4+ 21a^2h^5+ 7ah^6+ h^7\]
and negate that :) the add a^7 and we get


\[= -7a^6h -21a^5h^2-35a^4h^3- 35a^3h^4\]\[=>- 21a^2h^5- 7ah^6- h^7\]

then we can factor out an h i spose

Answer 6

\[-7a^6 -21a^5h-35a^4h^2- 35a^3h^3\]\[=>- 21a^2h^4- 7ah^5- h^6\]

and everthing with an h goes away at h=0 to leave us with:

-7a^6 up top

Answer 7

to fit it all back together we got:
\[\frac{-7a^6}{(a+0)^7}=\frac{-7a^6}{a^7}=\frac{-7}{a}\]
and im pretty sure i messed something

Answer 8

\[\frac{d}{dx}x^{-7}=-7x^{-8}=\frac{-7}{x^8}\]

Answer 9

so what am i missing :)

Answer 10

i forgot to include my a^7 in the common denominator after a combined the tops

Answer 11

do you see that?

Answer 12

\[\lim_{h->0}\frac{\cfrac{a^7-(a+h)^7}{a^7(a+h)^7}}{h}\]
\[\lim_{h->0}\frac{a^7-(a+h)^7}{a^7(a+h)^7h}\]
\[\frac{-7a^6}{a^7(a)^7}\]
\[\frac{-7}{a^7(a)}\]
\[\frac{-7}{a^8};\ and\ a=x\]

Answer 13

u were helping me in 2 seperate problems i feel silly now for replying to the wrong one