Find the point(s) (x,Y) at which the tangent line to x^2+4y+22=y^2+10x is horizontal.
What does it means when the tangent line is horizontal? what does the first derivative has to do with the "tangent"?
Differentiating this equation implicitly with respect to x yields 2x-10/2y-4. Thus we see that 2x-10=0 (2y-4 can't equal zero, that would be undefined) From here x=5. So the point is P(5,y). set x=5 in the original equation, solve for y and there you got the y. This turns out that that y has two values for x=5, and those being 1 and 3.
Btw, tangent comes from the latin word tangens, which means touch. So a tangent line touches a curve at just one point. And the derivative of the curve at that point gives us the slope of the tangent line.
Separate names with a comma.