Given a $ C^{\infty} $ function $ f:R^{2} \to R $ with $f(a,b)=0$. Suppose that $\left.\frac{df}{dy} \right| _{(a,b)}≠0 $ The implicit function theorem states that the level set $\{{(x,y):f(x,y)=0}\}$ is the graph of a smooth function $y=g(x)$ near $(x,y)=(a,b)$
I want to compute $ \left.\frac{dg}{dx} \right|_{a} $ and $ \left.\frac{d^2g}{dx^2}\right|_{a} $
I am studying for an exam. I have trouble understanding and solving the problems which are related to Implicit Function Theorem. How can we solve this problem? I need some more problems related to this theorem. Thanks in advance.