I'm reading this book and I'm stuck on page 223. Since some pages are missing, let me describe the problem formulation.
Preliminaries: Let $f:\mathbb R^n\to\mathbb R^n$ be $C^1$ vector field and consider an ODE $$ \dot x = f(x)\quad (1) $$ with a correspondent flow $\phi(t,x)$ such that $$ \frac{\partial \phi}{\partial t}(t,x) = f(\phi(t,x)) $$ and $\phi(0,x) = x$. Furthermore, let $$ Dg = \left(\frac{\partial g_i}{\partial x_j}\right)_{i,j=1}^n $$ denote the Jacobian matrix of $g:\mathbb R^n\to\mathbb R^n$.
Question: the matrix $H(t,x)$ is given by $H(t,x) = D\phi(t,x)$ where the Jacobian is w.r.t. $x$ coordinates only. It's written on the top of p. 223 that $$ \frac{\partial H}{\partial t}(t,x) = Df(\phi(t,x))H(t,x) $$ and I cannot understand why do we have $H(t,x)$ multiplier.
My thoughts are the following: $$\begin{split} \frac{\partial H_{ij}}{\partial t}(t,x) &= \frac{\partial}{\partial t}\frac{\partial \phi_{i}}{\partial x_j}(t,x) = \frac{\partial }{\partial x_j}\frac{\partial \phi_i}{\partial t} = \frac{\partial}{\partial x_j}f_i(\phi(t,x))\\&=\sum_{k=1}^n\frac{\partial f_i}{\partial \phi_j}(\phi(t,x))\frac{\partial \phi_k}{\partial x_j}(t,x) \end{split}$$
Are my calculations correct?