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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)

Guy L. A Level Further Mathematics  tutor, A Level Maths tutor, A Lev...

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