Function $ f(x,y)=x+\frac{y^3}{3} $ cut the xy-plane in a cutting-curve $h$. Find the tangent fuction of h in point$ P(9,-3)$.

Question

$$ f(x,y)=x+\frac{y^3}{3} , D(f)=\text{(x,y)}\in R |(x^2 +y^2\le2) $$

So what i thinking here is find the function h, then find the tangent. But i dont know how to do or is there another way? Thank you for reading

Answer 1

So you want to find the tangent of 3x + y$^3$ = 0 at (9,-3).
As 3 + 3y$^2$y' = 0, y'(x) = -/y$^2$.
At the given point, y'(9) = -1/9 = m.
The tangent is y + 3 = m(x - 9).