Let $f : \mathbb{R}^3 \to \mathbb{R}^3, f(x,y,z) = (x^2+y,y^3+2z,z^2)$. I have to find how does the differential of $f$ restricted to the tangent plane to the unit sphere $S^1$ in an arbitrary point $(x_0,y_0,z_0) \in S^1$ look.
I've done some work and I've found that the equation of the tangent plane to the unit sphere $S^1$ in an arbitrary point $(x_0,y_0,z_0) \in S^1$ is $$x_0x+y_0y+z_0z = 1$$ and the differential of $f$ in general is given by the Jacobian matrix, but I don't know how to find the form of $Df$ restricted to the plane above. Can someone give me a hint, please?