Basic partial derivative calculus and Ito Calculus

Question

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, after some statements about processes, it says that

Now I am not getting how this is resulted. Can anyone explain this? This seems to be simple matters.

Answer 1

This has nothing to do with stochastic calculus and is (ordinary) differential calculus: assume that $y=\varphi(x)$ and consider some function $u:y\mapsto u(y)$, thus $u(y)=(u\circ\varphi)(x)$. Differentiating this yields $$ \frac{\partial}{\partial x} u(y)=\varphi'(x)\cdot(u'\circ\varphi)(x)=\frac{\partial y}{\partial x}\cdot\frac{\partial u}{\partial y}. $$ Now, use $\varphi(x)=x^2$ on $x\gt0$, then $\varphi'(x)=2x=2\sqrt{y}$.