Help please!?

Question

A hyperbola has the equation y^3 + 3yx - x^3 = 9

-->By implicit differentiation w.r.t x, obtain the relation satisfied by dy/dx

-->Compute the coordinates of every point (if any) on the hyperbola at which the
tangent to the hyperbola is parallel to the line y=3x+1

Answer 1

The curve equation y³ + 3yx - x³ = 9

Differentiate implicitly with respect to x.

3y²(dy/dx) + 3[y + x(dy/dx)] - 3x² = 0

3y²(dy/dx) + 3y + 3x(dy/dx)] = 3x²

(dy/dx)(3y² + 3x) = 3x² - 3y

dy/dx = 3(x²- y)/3(y² + x)

dy/dx = (x²- y)/(y² + x)