Chain rule for the matrix function

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?

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The fact that $H$ comes up again on the RHS means that you have successfully linearised your initially nonlinear differential equation. So by "pretending" that $\phi$ and $H$ are independent quantities (when in reality, they are highly dependent on each other), you can use techniques of linear differential equations to get a lot of information on $H$, provided some small bits of information is assumed to be true for $\phi$.
That you can "assume a little and gain a lot more" is what then powers the analysis of the actual solution $\phi$.
Thanks, I'm familiar with Floquet's Theory - but apparently hasn't calculated derivatives for a long time. Namely, I realized the place when the chain rule appears only while typing the question - before I was missing the point that we should take derivative of $f$ w.r.t. $\phi_k$ rather that w.r.t. $x_k$. This fact always bothers me in partial derivatives of compositions – Ilya Dec 20 '11 at 15:12