I have a question:
Consider a function $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is differentiable. Find the derivative of the function: $G(x)=[ g\left ( x,x^{2},...,x^{n} \right )]^{2}$ where $x\in \mathbb{R}$.
Here, G is a function of one variable, so I tried to apply the chain rule to find its derivative as follows: G^{'}\left ( x \right )=2H^{'}\left ( x \right )H(x) where $H\left ( u_{1},u_{2},...,u_{n} \right )=g\left ( x,x^{2},...,x^{n} \right )$. Now to find the derivative of $H$, I did the following: $\frac{d}{dx}H=\frac{\partial H}{\partial u_{1}}\frac{\partial u_{1}}{\partial x}+...\frac{\partial H}{\partial u_{n}}\frac{\partial u_{n}}{\partial x}.$ Does what I did make sense?