# 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?