vector analysis – Finding the composition of multivariable functions and the Jacobian


Let $f: Bbb R^2to Bbb R^3, f(x) = (x_1,x_1x_2, x_2)$ , $g : Bbb R^2 to Bbb R, g(x) = |x|^2$ and let $h:= gf:Bbb R^2 to Bbb R^3.$ Show that $h$ is differentiable and find $Dh(x_0)$ at $x_0=(1,-1).$

So we need to find the Jacobian of the composition $gf$ and then see if the partials are continuous to determine if it’s differentiable and afterwards evaluate it at $(1,-1)$? This feels a bit weird since if I can find the Jacobian isn’t the function already differentiable?

Also the thing I’m stuck is that how do I find the composition of these kind of vector-valued functions?