Question

Find the derivative

Answer 1

\[f(x)=\sqrt{\left( \frac{ x }{ x^2+3 } \right)^{\arcsin(x^3)}+1}\]

Answer 2

It's a nasty one.

Answer 3

yikes 0_o

Answer 4

What I did was natural log both sides, but then I get stuck as I can't proceed with the logarithmic differentiation.

Answer 5

\[\ln(y)=\frac{ 1 }{ 2 }\ln \left( \left( \frac{ x }{ x^2+3 } \right)^{\arcsin(x^3)} +1\right)\]

Answer 6

Now what?

Answer 7

Hmm this might seem strange,
but I would recommend isolating the big chunk of x stuff before logging,
because ya, that +1 is causing a problem.

Answer 8

\[\large\rm y^2-1=\left(\frac{x}{x^2+3}\right)^{\arcsin(x^3)}\]

Answer 9

Wolfram wont even use this method.

Answer 10

Ohh okay so you're manipulating the inside. So what?

Answer 11

I'm just thinking that logging from THAT point might be easier:\[\large\rm \ln(y^2-1)=\arcsin(x^3)\cdot \ln\left(\frac{x}{x^2+3}\right)\]Hmm

Answer 12

How is that equivalent?

Answer 13

hmm? :o

Answer 14

Like how is what I had before and what you just got equivalent?

Answer 15

Did you not see the steps I took? :o I can explain them again maybe :D

Answer 16

Now I perfectly get what you did but how is what I had and what you had equivalent?

Answer 17

Hmm that question is confusing...
All I did was algebraic manipulation, that shouldn't change anything.

Answer 18

So ln(y^2) = ln(y^2-1)?

Answer 19

no?

Answer 20

That is what you wrote though?

Answer 21

\[\large\rm \ln(y)=\frac{1}{2}\ln \left(\left( \frac{x}{x^2+3}\right)^{\arcsin(x^3)} +1\right)\qquad \text{your equation}\]\[\large\rm \ln(y^2-1)= \ln\left[\left(\frac{x}{x^2+3}\right)^{\arcsin(x^3)}\right]\qquad \text{my equation}\]You see that the right side is missing the +1, yes?

Answer 22

I do. I suppose I am struggling to see the equivalence because I'm used to seeing it without logs.

Answer 23

Well, if you ignore the logs for a sec,
you understand how I went from here,\[\large\rm y=\sqrt{\left(\frac{x}{x^2+3}\right)^{\arcsin(x^3)}+1}\]to here?\[\large\rm y^2-1=\left(\frac{x}{x^2+3}\right)^{\arcsin(x^3)}\]

Answer 24

Yep I got that.

Answer 25

Ohh then you just logged both sides. Gotcha.

Answer 26

ooooo. K I get this now.

Answer 27

Thanks!

Answer 28

>.< heh

Answer 29

:)