## mathcalculus 2 years ago can someone explain this simple derivative to me please: sort(6x)

1. mathcalculus

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

2. mathcalculus

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

3. Psymon

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. Psymon

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

5. mathcalculus

ok

6. mathcalculus

im no sure about the chain rule its confusing

7. mathcalculus

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

8. Psymon

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. mathcalculus

no

10. mathcalculus

oh wait, i think so a while ago.

11. mathcalculus

let me open

12. Psymon

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. mathcalculus

does it always work?

14. Psymon

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. mathcalculus

thanks im going to read it right now

16. Psymon

Yeah, just let me know.

17. mathcalculus

ugh i gave it a shot. wrong. :(

18. Psymon

How'd it become 1/4?

19. mathcalculus

i did 1/2 *-1/2

20. Psymon

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. mathcalculus

huh?

22. Psymon

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. mathcalculus

down in the denominator?

24. mathcalculus

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

25. Psymon

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. mathcalculus

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

27. Psymon

It's the same thing actually xD

28. Psymon

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. mathcalculus

alrigght thanks ! :)))

30. Psymon

Mhm, np ^_^ Hope that made sense xD