Consider the curve defined by the equation x2y2 − 2x = 4 − 4y. Use implicit differentiation to find dy and write the equation of the tangent line at the point (2,2) in dx slope-intercept form.

Answer 1

(x²)(y²)-2x=4-4y ... (x²)(2ydy/dx)+(y²)(2x)-2=-4dy/dx ... dy/dx(2x²y+4)=2-2xy²

dy/dx= (2-2xy²)/( 2x²y+4) ... at P(2,2) dy/dx = (2-2*2*4)/(2*4*2+4)= -14/10

tangent line equation is ... y=mx + b where m is the slope and is equal to -14/10 found above. This y=-14/10x+b ... This line must by definition intersect with P. Using P(2,2), 2=-14/10*2+b finding b = 34/10.

Thus the tangent line is y=-14/10x +34/10