## anonymous 3 years ago can someone explain this simple derivative to me please: sort(6x)

1. anonymous

isn't it 1/2(6x)^(-1/2)

2. anonymous

how does it go from there to the answer: sqrt(6)/2 sqrt(x)

3. anonymous

It's chain rule so you also need to multiply by the derivative of the inside. So it would be what you put, but then multiply by 6.

4. anonymous

$\sqrt{6x} = \frac{ 1 }{ 2 }(6x)^{\frac{ -1 }{ 2 }}(6)$

5. anonymous

ok

6. anonymous

im no sure about the chain rule its confusing

7. anonymous

i know its: [f(g(x))]'= f'(g(x))*g'(x)

8. anonymous

Well, I have a unique way of showing it, so maybe it will help maybe not. Were you one of the ones I sent the derivative files to?

9. anonymous

no

10. anonymous

oh wait, i think so a while ago.

11. anonymous

let me open

12. anonymous

Ah xD Yeah, one of them had the chain rule in it. I teach it in a unique way, so maybe it helps maybe it doesn't.

13. anonymous

does it always work?

14. anonymous

Yeah. As long as you recognize what layers you have then yeah. Chain rule is multiplying the derivatives of each layer you have, making sure to not disturb what was originally inside of each layer.

15. anonymous

thanks im going to read it right now

16. anonymous

Yeah, just let me know.

17. anonymous

ugh i gave it a shot. wrong. :(

18. anonymous

How'd it become 1/4?

19. anonymous

i did 1/2 *-1/2

20. anonymous

Well, you bring the power down and then you made the power become -1/2. You're done with that layer, now you just go to the inner layer. $\frac{ 1 }{ 2 }(----)^{\frac{ -1 }{ 2 }}$ Thats it, no more to that part.

21. anonymous

huh?

22. anonymous

Well, when you do the outer layer. All you do is bring the power down as a multiplication then lower the power by 1. After that you are done with that layer. There's no other 1/2 to multiply by.

23. anonymous

down in the denominator?

24. anonymous

can you show me the steps to the answer? just want to see how you did it so i can ask

25. anonymous

Alright, so we have two layers. The first layer is simply ( )^1/2 and the inner layer 6x. So the chain rule says that we take the derivative of each layer and multiply the results. So following the normal derivative rule of $\frac{ d }{ dx }k ^{n} = nk ^{n-1}$, I'll do that with the first layer. $\frac{ 1 }{ 2 }(---)^{-\frac{ 1 }{ 2 }}$I just left the inner part blank for now, but that is the derivative of the first layer. Now I do the second layer, which is just 6x. So the derivativeof 6x is simply 6. So now that I have the derivative of both layers, I now multiply both of these derivatives $\frac{ 1 }{ 2 }(---)^{-\frac{ 1 }{ 2 }}*(6)$ This of course becomes: $\frac{ 3 }{ (---)^{\frac{ 1 }{ 2 }}}$ Now all that is left to do is plug back in what was originally inside of that layer, giving us: $\frac{ 3 }{ \sqrt{6x} }$

26. anonymous

oh the answer is sqrt (6)/2 sqrt(x)

27. anonymous

It's the same thing actually xD

28. anonymous

I'll show ya why: $\frac{ 3 }{ \sqrt{6x} }=\frac{ 3 }{ (\sqrt{6})(\sqrt{x)} }$Now multiply top and bottom by sqrt(6) $\frac{ 3(\sqrt{6)} }{ (\sqrt{6})(\sqrt{x})(\sqrt{6}) }$This then becomes finally: $\frac{ \sqrt{6} }{ 2\sqrt{x} }$

29. anonymous

alrigght thanks ! :)))

30. anonymous

Mhm, np ^_^ Hope that made sense xD