Question

Let f(x)=x^(1/2). Form and simplify the difference quotient [f(x+h) + f(x)]/h.

Answer 1

these problems are almost always about rationalizing the fraction

Answer 2

i.e. whenever you are trying to use the difference quotient on an expression under a radical it pays to try this:\[{f(x-h)-f(x)\over h}={\sqrt{x-h}-\sqrt x\over h}={\sqrt{x-h}-\sqrt x\over h}\cdot{\sqrt{x-h}+\sqrt x\over \sqrt{x-h}+\sqrt x}\]

Answer 3

Would I multiply both the numerator and the denominator by 1/(x+h)^1/2 +1/x^1/2 or by (x+h)^1/2 +x^1/2

Answer 4

*sorry that should all be +h in the above
a contagious typo from your post :P\[{f(x+h)-f(x)\over h}={\sqrt{x+h}-\sqrt x\over h}={\sqrt{x+h}-\sqrt x\over h}\cdot{\sqrt{x+h}+\sqrt x\over \sqrt{x+h}+\sqrt x}\]

Answer 5

you multiply by the conjugate of the numerator
the conjugate of a radical\[\sqrt a\pm\sqrt b\]is\[\sqrt a\mp\sqrt b\]

Answer 6

so because in the top we have \(\sqrt{x+h}-\sqrt x\) we need to multiply it by the its conjugate, which is \(\sqrt{x+h}+\sqrt x\)

of course, that means we need to multiply the bottom by that as well
this trick is called "rationalization"

Answer 7

what do you get in the numerator after multiplying by \(\sqrt{x+h}+\sqrt x\) ?

Answer 8

1/(x+h) - 1/x

Answer 9

just do the numerator\[(\sqrt{x+h}-\sqrt x)(\sqrt{x+h}+\sqrt x)=?\]

Answer 10

\[(x+h) - h\]

Answer 11

er I'm sorry (x+h) - x which would jsut be h

Answer 12

correct, which simplifies to just \(h\)
so \(h\) is in the numerator, and in the denominator is what?

Answer 13

\[h \sqrt{x+h} + h \sqrt{x} \]

Answer 14

yes
so can we cancel anything?

Answer 15

factor out the h

Answer 16

well, cancel out the h
we got one on top and bottom so it's gone

Answer 17

\[1/\sqrt{x+h} + \sqrt{x}\]

Answer 18

yeah, which we can take the limit of now :)

Answer 19

you should really have put parentheses around the denominator, but I know what you mean

Answer 20

Yeah sorry, first time using this site, still getting use to the equation maker

Answer 21

it's all good, and welcome to OS :)
so the limit of the above as h goes to zero is...?

Answer 22

\[1/(2 \sqrt{x})\]

Answer 23

right, which if you know the power rule you know is the correct answer

Answer 24

Haha, yeah, the power rule makes everything easier. Lots of arithmetic on that problem. Would that count as the simplified version?

Answer 25

the power rule has to be proven... using it is like a shortcut, so when they ask for the difference quotient you should do it if you have to show your work

you can always check your final answer with the power rule though

Answer 26

Awesome, thanks very much!

Answer 27

welcome :)