If x is a coordinate in the domain of g, and y is a coordinate in the range of f, how does y change with respect to x? This is an application of the chain rule presented earlier. Take the gradient of the component function fy and combine it with column x from g. If this is done for every x and y, the result is matrix multiplication.
Each entry in the resulting matrix is a good linear approximation to the change in the output variable, in response to the input variable. The rows of the resulting matrix are good linear approximations for the component functions, as they react to the input variables, and the entire matrix is a good linear approximation to f(g). Hence the composition is differentiable, and the derivative is f′(g)*g′.