Question

Find the equation of the tangent line:

Answer 1

\[x^2+y^2+\ln(xy)=2x\]at (1,1).

I've done the implicit differentiation as:
\[2x+2y(\frac{dy}{dx})+\frac{1}{xy}(y+x \frac{dy}{dx})=2\]But now I'm unable to figure out how to isolate (dy/dx). The most I've been able to simplify it is to the following:
\[2y(\frac{dy}{dx})+\frac{y+x \frac{dy}{dx}}{xy}=2-2x\]How can I proceed in isolating y'?

Answer 2

isolate dy/dx ? , lets make that equal w
so we have
2yw+(y+xw)/xy = 2-2x
muliply both sides by xy
2ywxy+y+xw=xy(2-2x)
w(2yyx+x)=xy(2-2x)-y
w=[xy(2-2x)-y]/(2yyx+x)

Answer 3

Yes, thank you. Sorry I didn't have a chance to reply until now. I was just seeing past the really simple solution. Thanks for the help!