I have the following equation:
$$\mu =\left [s_{\pi }-v(s_{\pi }-s_{W})+\zeta \right ]$$
And I have to derive its growth rate, which is:
$$\dot \mu =-\frac{v}{\mu } (s_{\pi }-s_{W})\dot v$$
Do you have any idea about the approach to follow? Thank you so much :)
First, note that the growth rate of $\mu$ is defined as $\dot\mu = \frac{ d\mu }{ \mu }$. Therefore, you will have to
For the first step, as in your previous question, simply use the rules of total differentials, specifically the product rule that states that for variables $X$ and $Y$, the total differential of $XY$ is given by
$$d(XY) = Y \, dX + X \, dY$$
Assuming that $s_\pi$, $s_W$ and $\zeta$ are (constant) parameters (so that by the constant rule, $ds_\pi = ds_W = d\zeta = 0$), we can apply this to your equation:
$$\begin{align} d\mu & = d \left[ s_\pi - v (s_\pi - s_W) + \zeta \right] \\ & = ds_\pi - d \left[ v (s_\pi - s_W) \right] + d\zeta \\ & = -\left[ (s_\pi - s_W) \, dv + v \, d(s_\pi - s_W) \right] \\ & = -(s_\pi - s_W) \, dv \end{align}$$
Now divide both sides by $\mu$, noting that since $\dot v = \frac{ dv }v$, we also have $dv = v \dot v$:
$$\begin{align} \dot\mu = \frac{ d\mu }\mu & = -\frac{ dv }\mu (s_\pi - s_W) \\ & = -\frac v\mu (s_\pi - s_W) \dot v \end{align}$$
which is the desired result. As before, see chapter 8 of Chiang and Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005, for more on total differentials. Alternatively, see this Q&A post of mine explaining the rules of total differentials and showing another example of how to apply them to an equation.
