How can I linearize the following equation (Bergman model)?

Question

I have to linearize the following equation so that I can use the Laplace transform and get the transfer function for the system. The equation is: $$\frac{dG(t)}{dt}=-p_1 G(t)-p_2 X(t)G(t)+ ....$$ $p_1$ and $p_2$ are problem parameters and $G(t)$ and $X(t)$ are the variables of the problem. I linearize $X(t)G(t)$ using the following Taylor approximation at a steady state $$X(t)G(t)\approx X_sG_s+\frac{\partial (X(t)G(t))}{\partial X(t)}|_s (X(t)-X_s)+\frac{\partial (X(t)G(t))}{\partial G(t)}|_s (G(t)-G_s)+...$$ Which yields $$X(t)G(t)\approx X_sG_s+G_s (X(t)-X_s)+X_s(G(t)-G_s)+..$$ Do you think that this is acceptable or should I instead take the partial derivatives with respect to time t?