Consider the polynomial
$$p(s;\tau,a)=3(56-56a\tau^2+a^2\tau^4)-112(-6+a\tau^2)\tau s-14(-36+a\tau^2)\tau^2s^2+112\tau^3s^3+7\tau^4 s^4,$$
I want to examine that how the roots of $p$ change with respect to the parameter $\tau$. I determined the derivative of a root with respect to $\tau$ using $$ \frac{\partial s}{\partial \tau}(s_i;\tau,a)=- \frac{ \frac{\partial p}{\partial \tau}(s_i;\tau,a)}{ \frac{\partial p}{\partial s}(s_i;\tau,a)},$$ where $s_i$ is one of the roots of $p$.
Is it correct to define the derivative $\frac{\partial s}{\partial \tau}$? And can we use the implicit function derivative formula?