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Question
Thank you for taking my question. I am homeschooling my son in Calculus; I've got my workbook, instructions, etc. and I haven't had much trouble before However, I just can't figure out today's lesson material, so I am having trouble teaching it. It has to deal with finding tangent lines to a curve. I was hoping you could show your answer to the problem below and then include your steps, hopefully a good example helps me make sense of it.

The question is:
Consider the curve x+xy+2y^(2)=6 Find the slope of the line tangent to the curve at the point (2,1)
Thanks again,
John

Answer
I'm not sure what section of the book you are in so let me give you two possible ways to do it. The first is to solve for y using the quadratic formula (the coefficients will have x's in them). Then take the derivative and plug in (2,1) to get the slope of the tangent line.

I suspect instead you are in the section in implicit differentiation, in which case you treat y as a function of x, even if you can't solve for y.  You apply d/dx to get:

1+y+xy' +4yy'=0.

Solve for y' to get y'=(-1-y)/(x+4y)

then plug in x=2, y=1 to get the slope which is -1/3. So the line is y-1=-1/3(x-2).

Teaching yourself implicit differentiation and then trying to explain it to your son is not going to be an easy task!

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