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Find the slope of the tangent line to the curve (a lemniscate) 2(x^2+y^2)^2 = 25(x^2-y^2) at point (-3,-1).

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just use implicit differentiation:

2(x^2+y^2)(2x+2yy') = 25(2x-2yy')
y' = -x(4x^2+4y^2-25) / y(4x^2+4y^2+25)
So, at (-3,-1)
y' = 3(15)/-(65) = -9/13

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