Question

if y^6=tanh^-1(tan^-4(4x^2+y^2)^2)^9
find dy/dx
show work....

Answer 1

this is just stupid.

y^6 = tanh^-1(u)^9
6y^5 y' = 9 tanh^-1(u)^8 * 1/(1-u^2) du/dx

where u = tan^-4(v^2)
and v = 4x^2+y^2
= cot^4(v^2)

du/dx = 4 cot^3(v^2) (-csc^2(v^2)) (2v dv/dx)

dv/dx = 8x + 2yy'

mash all that together and solve for y'

I see little to be gained from such an artificial jumble.

Answer 2

Here's what wolframalpha.com ends up with:

http://www.wolframalpha.com/input/?i=derivative+y%5E6%3Dtanh%5E-1(tan%5E-4(4x%5E2%2By%5E2)%5E2)%5E9

That's as you typed it. I interpreted it as

http://www.wolframalpha.com/input/?i=derivative+y%5E6%3Dtanh%5E-1(tan%5E-4((4x%5E2%2By%5E2)%5E2))%5E9

which is even messier