I am currently doing chain rule exercises and I encountered this one:
Calculate the derivative of $g(f(x,y))$ at $(1,0)$:
\begin{align*} f: \mathbb{R}^2 \to \mathbb{R}^3, f(x,y) &= (x^2 +\sin (y), xy, \sin(xy))\\ g: \mathbb{R}^3 \to \mathbb{R}^2, g(x,y,z) &= (xy^2+z, \frac{1}{x^2 + z^2}) \end{align*}
By 'the derivative' I imagine they mean the directional derivative at $(0,1)$ given by a general $(x,y)$ vector. My question lies with how do we process this derivative? I've done other examples but none as confusing as this one.