Question

using the definition of a derivative find f(x)
=-2/sqrt{x}

Answer 1

\[f(x)= -2/ \sqrt{x}\]

Answer 2

this will be a pain to write out but not hard to compute. you must use the definition yes?

Answer 3

i know that part i am having problems with rationalizing the denominator

Answer 4

you need to compute
\[lim_{h->0}\frac{\frac{-2}{\sqrt{x+h}}+\frac{2}{\sqrt{x}}}{h}\]

Answer 5

Good opportunity to practice your LaTeX satellite! :)

Answer 6

go ahead watchmath.

Answer 7

i am learning. eventually i will be fluent in it

Answer 8

i know that part the step after that

Answer 9

ok lets give me a break and at least factor out the -2, since this has nothing to do with the limit ok?

Answer 10

just factor out by 2 thats all you had t say lol thanks so much

Answer 11

i knew that i was just doubting myself

Answer 12

i knew that i was just doubting myself

Answer 13

so we will just write \[lim_{h->0}\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}\]

Answer 14

all divided by h of course. we worry about that last.

Answer 15

so now we rationalize

Answer 16

\[\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}=\frac{\sqrt{x}-\sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}\]

Answer 17

yes multiply numerator and denominator by the conjugate of th enumerator.

Answer 18

which of course is \[\sqrt{x}+\sqrt{x+h}\]

Answer 19

when you do that the numerator will just be
\[x-x+h=h\]

Answer 20

and the denominator will be the product
\[\sqrt{x} \sqrt{x+h}) (\sqrt{x}+\sqrt{x+h})\]

Answer 21

sorry am not repsonding i was working it out thanks i got it

Answer 22

thats ok i was typing. did you get the numerator correctly? it is just h

Answer 23

yes the x's cancel

Answer 24

the denominator is that ugly thing i wrote last. so you have in total
\[\frac{h}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x}+\sqrt{x+h}}\]

Answer 25

yes they 'cancel' meaning they add to zero. and of course dividing by h means the h goes in the denominator

Answer 26

the h cancels

Answer 27

so that last ugly thing i wrote is what you get when you rationalize the numerator. yes now the h cancels.

Answer 28

leaving
\[\frac{1}{(\sqrt{x}\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})}\]

Answer 29

now you can replace h by 0 since you will not be dividing by 0.

Answer 30

to get...
\[\frac{1}{(\sqrt{x}\sqrt{x})(\sqrt{x}+\sqrt{x})}\]

Answer 31

yes

Answer 32

\[=\frac{1}{2x\sqrt{x}}\]

Answer 33

oh damn i made a mistake early on
the numerator was \[x-(x+h)=-h\]
not \[x-x+h\]

Answer 34

a very bush league mistake but easily rectified. just replace the 1 in the numerator by -1

Answer 35

sorry!

Answer 36

so the correct answer is
\[\frac{-1}{2x\sqrt{x}}\]

Answer 37

then don't forget to multiply by the -2 at the end because we factored it out.

Answer 38

so the "final answer" as we say is
\[\frac{1}{x\sqrt{x}}\] sorry it took a while

Answer 39

all steps clear?

Answer 40

sorry my computer is freezing idk if its thes site or what

Answer 41

sometime the site is funky. earlier today it certainly was.

Answer 42

if you have a question about any step post and i will respond

Answer 43

i understand thank you sooo much!