slope of the tangent line

Question

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

Answer 1

First, just as a sanity check, verify that x=-3, y=1 is a solution:

  2(9+1)² =?= 25(9-1)

    2*100 =?= 25*8

Yes, both sides are 200.

 

OK, now differentiate each side with respect to x:

 d/dx(    2(x²+y²)² = 25(x²-y²)   )

 

   2 * 2(x²+y²)*(2x+2y*y')  = 25(2x -2y*y')

 

where y' is dy/dx = the slope of the tangent line.

 

Put in the known values x=-3, y=1 and solve for y' :

    

         [excersize for reader]

  

    y' = 9/13  si the slope at the point (-3,1), hence the  slope of the tangent line.