Let $f(x,y,z) = (xyz,x^2z^2,y^3x)$ and $g(x,y,z) = (x+y, yz, z-2x, x^2y)$.
I need to use the chain rule to find D(g $\circ$ f) which means I have to compute $\frac{\partial g_1}{\partial f_1}$ $\frac{\partial f_1}{\partial x}$ but $f_1$ consists of $x+y$ and I am not sure how to compute $\frac{\partial g_1}{\partial f_1}$ when $f_1$ is not a single variable.
For the second part, I have to compute g $\cdot$ f and D(g $\cdot$ f) but f $\in \mathbb{R^3}$ and g $\in \mathbb{R^4}$ so the dot product is not even possible.
What am I missing here?
Thanks!