SOLUTION: Hello good friends . I have an implicit differentiation question if someone could please be kind enough to help me out . implicitly differentiate , ycos(1/y)=4x+4y they are asking

Algebra ->  Finance -> SOLUTION: Hello good friends . I have an implicit differentiation question if someone could please be kind enough to help me out . implicitly differentiate , ycos(1/y)=4x+4y they are asking      Log On


   

Question 958977: Hello good friends . I have an implicit differentiation question if someone could please be kind enough to help me out . implicitly differentiate , ycos(1/y)=4x+4y they are asking for dy/dx thank you! this is a tough one ! At least for me : )
Found 2 solutions by stanbon, Fombitz:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
(1/y)=4x+4y
----
Rewrite::
y^-1 = 4x + 4y
----
Differentiate::
-y^-2*y' = 4 + 4y'
----
solve for y'
6y' = -y^-2 - 4
----
y' = (-1/6)((1/y^2)+4)
==================
Cheers,
Stan H.
------------

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Step by step.
It's not tough, but it's easy to make a mistake.
Left side, you have a product.
So the derivative a product is,
%28d%28ab%29%29%2Fdu=a%28db%2Fdu%29%2Bb%28da%2Fdu%29
So left hand side is,
y%2Ad%28cos%281%2Fy%29%29%2Fdy%2Bcos%281%2Fy%29%2A%281%29
Use the chain rule for the first derivative,
d%28cos%281%2Fy%29%29%2Fdy=-sin%281%2Fy%29%2A%28-y%5E%28-2%29%29
d%28cos%281%2Fy%29%29%2Fdy=sin%281%2Fy%29%2F%28y%5E%282%29%29
Putting it all together,
%28sin%281%2Fy%29%2Fy%2Bcos%281%2Fy%29%29dy
Now the right hand side is straight forward,
4dx%2B4dy
So then,
%28sin%281%2Fy%29%2Fy%2Bcos%281%2Fy%29%29dy=4dx%2B4dy
%28sin%281%2Fy%29%2Fy%2Bcos%281%2Fy%29-4%29dy=4dx
dy%2Fdx=4%2F%28cos%281%2Fy%29-sin%281%2Fy%29%2Fy-4%29
Re-arranging,
highlight%28dy%2Fdx=4y%2F%28y%28cos%281%2Fy%29-4%29-sin%281%2Fy%29%29%29