Question

find the derivative of f(x)=x/sqrtx in 2 different ways

Answer 1

Hmm ok.
So there are several ways to approach this.
Have you tried anything yet? :o

Answer 2

One way is the quotient rule.

Answer 3

use the quotient rule and the product rule

Answer 4

basically the quotient rule can be derive using the product rule

Answer 5

derived*

Answer 6

\[\frac{ x }{ \sqrt x } = (\frac{ x }{ \sqrt x } ) \frac{ \sqrt x }{ \sqrt x } = \sqrt x\]

Answer 7

product rule's general form is

f'(x) = u'v + uv'

when applying a quotient into the product rule
u/v
you will end up with
u * v^-1

u(x) * [v(x)]^-1

Answer 8

what on earth are you doing @jayz657

Answer 9

it can be simplified into sqrt(x) and you can just take the derivative of sqrt(x) using the power rule

Answer 10

True it does.
I think he was commenting on the steps you showed.
They're really strange :) lol

Answer 11

i multiplied by 1 since sqrt(x)/sqrt(x) = 1

Answer 12

you'll end up using the power rule when you derive the product rule

Answer 13

Oh ok, so I guess there was another step in there, cancelling the x's.
I see :o

Answer 14

power rule and chain rule in fact

Answer 15

are you suggesting that f'(x) = sqrt (x) is the same as f'(x) = x/sqrt(x) ?

Answer 16

No. Jay was simplifying the problem before taking the derivative.

He was asked to show several ways to take the derivative, so that seems like one nice approach.

Answer 17

yea i rationalized the denominator and it simplified to sqrt(x)

Answer 18

you're still stuck with deriving the same problem utilizing 2 different techniques
here are the following:
by the definition
by product rule (needs chain and power rules)
by quotient rule

Answer 19

\[\Large (\sqrt x)'\quad=\quad (x^{1/2})'\]Why would you need product or quotient rule to find that derivative? :\
Maybe I'm misunderstanding what you're saying..
You just use power rule for that one.

Answer 20

i would prob just use the quotient rule as the 2nd method

Answer 21

it is 1/sqrt(x) in this case
which is why I said in my earlier post

product rule's general form is

f'(x) = u'v + uv'

when applying a quotient into the product rule
u/v
you will end up with
u * v^-1

u(x) * [v(x)]^-1

Answer 22

ya, 1/(2sqrt(x))
:o

Answer 23

I like how the asker hasn't replied at all LOL

Answer 24

either method works fine

Answer 25

XD

Answer 26

I know they all work fine
the purpose of the exercise is to find out one's proficiency at using all derivative techniques