Use implicit differentiation to find an equation of the tangent line?

First, take the derivative of the function so
2x+2y+2xy'-2yy'+1=0  Note, you have to use the product rule when you take the derivative of 2xy. Also, when you take the derivative of y, you're left with dy/dx which is the same as y'. When you take the derivative of x, you're left with dx/dx, which is just 1, so we don't have to write anything like with y'.

Next, isolate y' so that you get y'=.... so
2xy'-2yy'= -2x-2y-1
y'(2x-2y)= -2x-2y-1
y'=(-2x-2y-1)/(2x-2y)

Now, plug in 5 for x and 6 for y
y'=-23/-2= 23/2 which is your slope.

y-y1=m(x-x1)
y-6=(23/2)(x-5)

take the derivative to find the slope.
2x+2y+2xdy/dx-2ydy/dx+1=0
dy/dx(2x+2y)=1-2x
dy/dx+(1-2x)/(2x+2y)
plug in 5 for x and 6 for y
the slope is   -9/22
use point slope form to find the equation

Ok, the first thing to do is to find the derivative:

The implicit differentiation method is to differentiate y as (dy / dx), and solve for dy / dx

First, derive the x's and constants, they are the easiest.

We have:

2x + (dy / dx)[2xy] - (dy / dx)[y ^ 2] + 1 = 0

Now, to differentiate 2xy, use the product rule. First, set 2 out:

(dy / dx)[2xy] = 2(dy / dx)[xy]

You get,

(dy / dx)[2xy] = 2(y + x * (dy / dx))

So far,

2x + 2(y + x * (dy / dx)) - (dy / dx)[y ^ 2] + 1 = 0

To differentiate (y ^ 2) use the power rule. You get,

(dy / dx)[y ^ 2] = 2y * (dy / dx)

We have,

2x + 2(y + x * (dy / dx)) - 2y * (dy / dx) + 1 = 0

NOTE: In this case, after differentiating, (dy / dx) is treated like a multiplicand.

Distribute through (this will make it easier to isolate (dy / dx).

2x + 2y + 2x(dy / dx) - 2y(dy / dx) + 1 = 0

Factor the (dy / dx) terms out.

2x + 2y - (2y - 2x)[dy / dx] + 1 = 0

Add (2y - 2x)[dy / dx] to both sides:

2x + 2y + 1 = (2y - 2x)[dy / dx]

Divide both sides by (2y - 2x) to isolate dy / dx.

dy / dx = (2x + 2y + 1) / (2y - 2x)

Now we have the derivative. Calculate the slope at the point (5, 6) by plugging in x and y.

dy / dx = (2(5) + 2(6) + 1) / (2(6) - 2(5))

dy / dx = (10 + 12 + 1) / (12 - 2)

dy / dx = 23 / 2

dy / dx = 11.5

The slope is 11.5

Now, just use the point slope form of the equation, using (5, 6) and the slope 11.5.

y - 6 = 11.5(x - 5)