x^2 + y^2 + 10x + 2y - 4xy = 10. Find dy/dx in terms of x and y, fully simplifying your answer.

x² + y² + 10x + 2y - 4xy = 10

Start by differentiating both sides by x, the terms not containing y are differentiated normally, x² becomes 2x, 10x becomes 10, and 10 becomes 0.
For the y² term, by implicit differentiation the result is 2y (the same as with x) multiplied by a factor of dy/dx, or 2y(dy/dx).
For the 2y term, as above, the result is the same as for x, 2 in this case, multiplied by a factor of dy/dx. I.e. 2(dy/dx).
The -4xy term requires use of the product rule (d(uv)/dx = v(du/dx) + u(dv/dx)). In this case that gives d(-4xy)/dx = -4x(dy/dx) + y(-4). 
Our completely differentiated equation is now
2x + 2y(dy/dx) + 10 + 2(dy/dx) -4x(dy/dx) - 4y = 0
Grouping the dy/dx terms and taking the remaining terms to the other side gives
(2y + 2 - 4x)(dy/dx) = 4y - 10 - 2x
Dividing through by (2y + 2 - 4x) gives
(dy/dx) = (4y - 10 - 2x)/(2y + 2 - 4x)
Simplifying the fraction on the Right Hand Side gives
(dy/dx) = (2y - 5 - x)/(y + 1 - 2x)