Show that there exists a differentiable function $y = g(x)$ defined in some neighborhood $(a, b)$ of $0$ that solves the equation $(y^2 + 2)\times\sin y = 2\times x^3$. I.e., $$(g(x)^2 + 2)\times \sin g(x) = 2\times x^3$$ It is satisfied for all $x$ from the domain of $g$ and $g(0) = 0$.
Determine the derivative of this function at $0$?
I know that it relates to the implicit function theorem but I do not know how to do this.
Hint:
Let $F$ be given by
$$F(x,y)=(y^2+2)\sin y-2x^3$$
By the implicit function theorem
$$g'(0)=-\frac{F_x(0,0)}{F_y(0,0)}$$
so long as $F_y(0,0)\neq 0$.
Alternatively, use the chain rule (and product rule) to differentiate both sides of
$$([g(x)]^2+2)\sin [g(x)]=2x^3$$
and then solve for $g'(x)$ and evaluate at $x=y=0$.