Question

Use first principles to find the derivative of h(x) = 2sqrtx

Answer 1

\[\lim _{ {h \rightarrow 0}} \frac{f(x+h)-f(x)}{h}\], if I'm not mistaken?

Answer 2

@pingpong21 you there?

Answer 3

Sorry, but yest that's correct

Answer 4

:)

Thanks!

Need some help with that limit?

Answer 5

Yeah, just how to get to the final answer

Answer 6

To get the derivative I mean

Answer 7

Sorry, I had some internet problems....
Could you get the answer in the meantime? or should I quickly help?

Answer 8

Could use some help if you can

Answer 9

No problem!

Okay, so \[\lim _{ {h \rightarrow 0}} \frac{f(x+h)-f(x)}{h} \\ =\lim _{ {h \rightarrow 0}} \frac{2\sqrt{x+h}-2\sqrt{x}}{h}\]
and that equals\[\lim _{ {h \rightarrow 0}} \frac{2}{\sqrt{x+h}+\sqrt{x}}\]

Answer 10

Now, by the properties of limits, we get\[=\frac{2}{\color {red}{(\lim _{ {h \rightarrow 0}}\sqrt{x+h})}+\sqrt{x}}\]

Answer 11

can you try to solve that?

Answer 12

Wouldn't it just disappear?
Given h goes to 0

Answer 13

and what rule allows you to drop the sqrt to the denominator?

Answer 14

Yes!
h goes to 0, so you can replace it with 0.

Answer 15

The fraction was rationalized, that is making it so that there are no sqrt's and stuff in the numerator

Answer 16

alright, thank you very much

Answer 17

No problem:)

Looks like you understand what to do?

Answer 18

Yup. Just drop the sqrt to the bottom if it shows

Answer 19

An easy way is also to write the roots as exponentials (e.g sqrt(x) = x^(1/2)).
However, in this specific case it wouldn't really have helped