Consider the following slowly time-varying system $$\epsilon\dot{z}(t)=A(x(t))z(t),$$ $$\dot{x}(t)=f(x(t)).$$ Under some stability assumptions, it is proven that there exist a Lyapunov function for the system of the form $$W(x,z)=z^\top P(x)z,$$ with $P(x)$ being a positive-definite symmetric matrix satisfying the equality $$P(x)A(x)+A^\top(x)P(x) = -I.$$ In particular, the explicit form of $P(x)$ is given by $$P(x)=\int_0^\infty e^{A^\top(x)\sigma}e^{A(x)\sigma}d\sigma.$$ At this point, it is given that the derivative of $W(x,z)$ along the trajectories of the system is: $$\dot{W}=-\frac{1}{\epsilon}z^\top z+z^\top\dot{P}(x)z.$$
However, I don't understand exactly how to retrieve this expression since I go about it as: $$\dot{W} = \frac{\partial W}{\partial x}\dot{x} + \frac{\partial W}{\partial z}\dot{z},$$ but then I don't know how to compute $\frac{\partial W}{\partial x}$ and $\frac{\partial W}{\partial z}$.
We can differentiate $\dot W$ directly without the chain rule. I'll omit all $t$ dependencies. As $W=z^TPz$ the derivative is obtained from the product rule $$ \dot W=\dot z^TPz+z^T\dot Pz+z^TP\dot z. $$ The second term is present in the final expression, so we combine the first and the last ones and replace $\dot z=\frac{1}{\epsilon}Az$ $$ \dot z^TPz+z^TP\dot z=z^T\left[\frac{1}{\epsilon}A^TP+P\frac{1}{\epsilon}A\right]z=\frac{1}{\epsilon}z^T[\underbrace{A^TP+PA}_{=-I}]z=-\frac{1}{\epsilon}z^Tz. $$