Question

Need help with understanding derivative for ((kx^3)^1/2

Answer 1

\[\sqrt{kx^3}\]

Answer 2

So i need help with finding the derivative of that.

Answer 3

\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{x}=\frac{d}{dx}x^{1/2}=\frac{1}{2}x^{(1/2)-1}=\frac{1}{2\displaystyle\sqrt{x}}}\)

Answer 4

Um actually no. According to Khan Academy is \[\frac{ 3 }{ 2}\sqrt{kx}\]

Answer 5

Unless that wasnt your final answer :o

Answer 6

oh nvm i get what your saying

Answer 7

If that answer is correct, I am pretty sure I can come up with this answer too:)

Answer 8

You do know, as I said that
\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{x}=\frac{1}{2\displaystyle\sqrt{x}}}\)

and thus, by the chain rule principle, in general you have:
\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{f(x)}=\frac{1}{2\displaystyle\sqrt{f(x)}}\times f'(x)=\frac{f'(x)}{2\displaystyle\sqrt{f(x)}}}\)

and thus, in your particular case you have:
\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{kx^3}=\frac{(kx^3)'}{2\displaystyle\sqrt{kx^3}}}\)

Answer 9

and this (if you differentiate correctly) does simplify to the result that you have according to khans academy.

Answer 10

(assuming \(x\in\mathbb{R}\) of course, because \(i^2\ne\displaystyle \sqrt{i^2}\) .)

Answer 11

Well this particular question was a "Power Rule Challenge" and it was before I would start learning about the chain rule. I'm just confused as to what do when I reach this point.
\[\frac{ d }{ dx }(k ^{1/2}x ^{3/2})\]

Answer 12

I could show all my steps before if you would like as well

Answer 13

Yes, indeed \(\color{black}{\displaystyle \sqrt{kx^3}=\displaystyle k^{1/2}x^{3/2}}\).

Answer 14

So, if you don't want to use the chain rule, you are also fine with that.
\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{kx^3}=\displaystyle \frac{d}{dx}k^{1/2}x^{3/2}=\left(k^{1/2}\right)\frac{d}{dx}x^{3/2}}\)

Answer 15

Yes

Answer 16

But, avoiding the chain rule is not a way out, because there are countless problems that you will very likely have to do for which Chain Rule Would be the ONLY way.

Answer 17

Yes I understand that. I'm actually moving on to learning that but this particular question has been bothering me.

Answer 18

but, now, this question is clear, isn't it?

Answer 19

Well, for the very least, you know how to differentiate \(k^{1/2}x^{3/2}\). Right?

Answer 20

So the next step it would be...
\[k ^{1/2}\frac{ d }{ dx }(\frac{ 3 }{ 2 }x ^{1/2})\]

Answer 21

oh i see my error looking at it now :o

Answer 22

you don't need `d/dx` once you have already differentiated.
and yes, \(\color{black}{\displaystyle \frac{d}{dx}(k^{1/2}x^{3/2})=\sqrt{k}\times (3/2)x^{1/2}=\frac{3}{2}\sqrt{kx}}\).

Answer 23

Yea silly me lol. I see it now :D

Answer 24

But what happens to the x^1/2?

Answer 25

\(\color{black}{\displaystyle x^{1/2}=\sqrt{x}}\)

Answer 26

Oh i see.
\[\sqrt{k}\sqrt{x}=\sqrt{kx}\]

Answer 27

Yup, and again assuming \(k>0\) and \(x>0\).

Answer 28

Just posting this "assumptions" if you like preciseness. (If not, just ignore them.)

Answer 29

oh I mean \(k,x\ge0\) .

Answer 30

Haha ok. Thank you so much for clearing this up for me. I really appreciate it :D

Answer 31

anyway,

if you were to do it with the chain rule, then
\(\color{black}{\displaystyle \frac{d}{dx} \sqrt{kx^3}=\frac{(kx^3)'}{2\displaystyle\sqrt{kx^3}}=\frac{3kx^2}{2\displaystyle\sqrt{kx^3}}=\frac{3kx^2}{2\displaystyle\sqrt{kx^2}\times \sqrt{x}}=\frac{3kx^2}{2\displaystyle\sqrt{{\tiny~~}k{\tiny~}}\color{blue}{x}\times \sqrt{x}}.}\)

This \(x\) in BLUE, would be \(\sqrt{x^2}=|x|\), but since \(x\ge0\) anyway,
because you have an \(x^3\) inside the square root which would demand such a condition.

Then,
\(\color{black}{\displaystyle \frac{3kx^2}{2\displaystyle\sqrt{{\tiny~~}k{\tiny~}}x \sqrt{x}}=\frac{3}{2}\times\frac{k}{\sqrt{k}}\times\frac{x^2}{\sqrt{x \sqrt{x}}}=\frac{3}{2}\sqrt{k}\times \sqrt{x}=\frac{3}{2}\sqrt{kx} }\)

Answer 32

Alright good to know. I'll be sure to look back at this after I learn the chain rule in a few :D

Answer 33

Good Luck!