Application of Derivative

Question

what is the slope of the tangent line to the curve x² + 2x² + y² +4y⁵ +2y = 1 at the point (1,0)?

Answer 1

x² + 2x² = 3x²

 

So, we have 3x² + y² + 4y⁵ + 2y = 1

 

Differentiate with respect to x:  6x + 2y(dy/dx) + 20y⁴(dy/dx) + 2(dy/dx) = 0

 

                                                      (dy/dx)[2y + 20y⁴ + 2] = -6x

 

Evaluate when x = 1 and y = 0:  (dy/dx)(2) = -6

 

                                                       dy/dx = -3 = slope of tangent line at (1,0)

 

Equation of tangent line:  y-0 = -3(x-1)

 

                                        y = -3x + 3