In Boyd's Convex Optimization text, on page 86, there is the following equation:
$$f(x) = h(g(x)) = h(g_1(x), \cdots, g_k(x))$$
Then, in order to show convexity, we take the second derivative, yielding:
$$f''(x) = g'(x)^T \nabla^2h(g(x))g'(x) + \nabla h(g(x))^T g''(x)$$, with $h:R^k \rightarrow R, g_i:R^n \rightarrow R$
There was a previous equation on page 71 which was similar, but yielded the gradient of the function f(x). This time, the second derivative is a single number. I was wondering how do I get the second derivative calculation? I know it is a derivative of some kind of a composition function. h() is a scalar function, that is, it takes in a vector and returns a real number. g() is a vector function, that is, it takes in a number and returns a vector.