Find the slope of the tangent line to the curve xy^3+y-1=0 at the point (1,0)

Answer 1

Use implicit differentiation, taking note that functions of y act as composite functions of x

xy³ + y -1 = 0

y³ +3xy²y' +y' =0

(3xy² +1)y' = -y³

y' = -y³/(3xy² +1)

To obtain the tangent line evaluate y' at (1,0) and set up the point slope equation.