Finding the derivative of a relational problem I am self studying some calculus and I have gotten really stuck! I thought I had the right idea but I keep getting the answer totally wrong. I am sure I am missing something important. Here is the problem:
For the equation $6x^{\frac{1}{2}}+12y^{-\frac{1}{2}} = 3xy$, find an equation of the tangent line at the point $(1, 4)$.
Here is my work:
$$6x^{\frac{1}{2}}+12y^{-\frac{1}{2}} = 3xy$$
$$6(x^{\frac{1}{2}}+2y^{-\frac{1}{2}}) = 3xy$$
$$2(x^{\frac{1}{2}}+2y^{-\frac{1}{2}}) = xy$$
$$2x^{\frac{1}{2}}+4y^{-\frac{1}{2}} = xy$$
$$\ln(2x^{\frac{1}{2}})+\ln(4y^{-\frac{1}{2}}) = \ln(xy)$$
$$\ln2 + \ln x^{\frac{1}{2}}+\ln4 + \ln y^{-\frac{1}{2}} = \ln x + \ln y$$
$$\ln2 + \frac{1}{2}\ln x+\ln4 -\frac{1}{2} \ln y = \ln x + \ln y$$
$$\ln2 + \ln4 + \frac{1}{2}\ln x - \ln x =  \ln y + \frac{1}{2} \ln y$$
$$\frac{2}{3}\ln2 + \frac{2}{3}\ln4 - \frac{1}{3}\ln x = \ln y$$
$$e^{\frac{2}{3}\ln2 + \frac{2}{3}\ln4 - \frac{1}{3}\ln x} = y$$
$$\frac{2}{3}\ln2 + \frac{2}{3}\ln4 - \frac{1}{3}\ln x = \ln y$$
$$\frac{1}{3x} = y'\frac{1}{y}$$
$$y' = \frac{y}{3x}$$
$$y' = \frac{e^{\frac{2}{3}\ln2 + \frac{2}{3}\ln4 - \frac{1}{3}\ln x}}{3x}$$
$$y - 4 = \frac{4}{3}(x-1)$$
$$y = \frac{4}{3}x-\frac{4}{3} + 4$$
$$y = \frac{4}{3}x+\frac{8}{3}$$
The Solution I found was: $y = \frac{4}{3}x+\frac{8}{3}$ but this is wrong! Can you tell me where I have gone wrong?

In general, you've just put yourself through a lot of unnecessary work.  Just hit the thing immediately with implicit differentiation:
$$6x^{\frac{1}{2}}+12y^{-\frac{1}{2}} = 3xy$$
Becomes:
$$3x^{-1/2}-6y^{-3/2} = 3xy' + 3y$$
You're told to find the tangent line at $(1, 4)$, so plug in $x=1,\;y=4$:
$$3(1)^{-1/2}-6(4)^{-3/2} = 3(1)y' + 3(4)$$
$$3-6\frac{1}{8} = 3y' + 12$$
Now solve for $y'$:
$$y' = \frac{-9-\frac{3}{4}}{3}$$
EDIT: Dang.  I missed the "easy" mistake, and found a harder one instead. :o
Doing it your way:
$$\frac{2}{3}\ln2 + \frac{2}{3}\ln4 - \frac{1}{3}\ln x = \ln y$$
When you differentiated, you forgot a minus sign:
$$\color{red}- \frac{1}{3x} = \frac{y'}{y}$$
