We have the following four analytic functions of a real variable:
\begin{align} \lambda (\epsilon) &= 0 + \lambda_1 \epsilon + \lambda_2 \epsilon^2 + \ldots \\ \tilde{\lambda} (\epsilon) &= 0 + \tilde{\lambda}_1 \epsilon + \tilde{\lambda}_2 \epsilon^2 + \ldots \\ \kappa (\epsilon) &= \kappa_0 + \kappa_1 \epsilon + \kappa_2 \epsilon^2 + \ldots \\ \tilde{\kappa} (\epsilon) &= \tilde{\kappa}_0 + \tilde{\kappa}_1 \epsilon + \tilde{\kappa}_2 \epsilon^2 + \ldots \end{align}
where $\kappa_0, \tilde{\kappa}_0 > 0$ and the expansions are valid for $0\leq \epsilon < R$, for some $R>0$. Now, we want to analyze a function of the form
$$ f(\epsilon) = -\lambda (\epsilon)\ln \lambda (\epsilon) +\tilde{\lambda} (\epsilon)\ln \tilde{\lambda} (\epsilon) -\kappa (\epsilon)\ln \kappa (\epsilon) +\tilde{\kappa} (\epsilon)\ln \tilde{\kappa} (\epsilon). $$
In particular, for sufficiently small $\epsilon$, we would like to "ignore" all the terms of order 2 or higher from the above expansions so as to get the following form for the derivative for $\epsilon > 0$:
$$ f'(\epsilon) = \ln (1/\epsilon) [\lambda_1 - \tilde{\lambda}_1 ] + Err(\epsilon), $$ where $Err(\epsilon)$ can be uniformly bounded as a function of $\epsilon$. Are we justified in making the above approximations. If yes, how can we give a mathematically rigorous reasoning for this justification?
Consider that this is the composition of Taylor series. Changing notations, you want to compute $$f(\epsilon)=-y_1(\epsilon)\log[y_1(\epsilon)]+y_2(\epsilon)\log[y_2(\epsilon)]-y_3(\epsilon)\log[y_3(\epsilon)]+y_4(\epsilon)\log[y_4(\epsilon)]$$ where $$y_1(\epsilon)=a_1\epsilon+a_2\epsilon^2+O(\epsilon^3)$$ $$y_2(\epsilon)=b_1\epsilon+b_2\epsilon^2+O(\epsilon^3)$$ $$y_3(\epsilon)=c_0+c_1\epsilon+c_2\epsilon^2+O(\epsilon^3)$$ $$y_4(\epsilon)=d_0+d_1\epsilon+d_2\epsilon^2+O(\epsilon^3)$$
This will give the expansion of $f(\epsilon)$ to $O(\epsilon^3)$ (I shall not type it).
Computing the derivative, we shall find $$f'(\epsilon)=K+(b_1-a_1) \log (\epsilon )+O\left(\epsilon ^1\right)$$ $$K=-a_1(1+\log(a_1))+b_1(1+\log(b_1))-c_1(1+\log(c_0))+d_1(1+\log(d_0))$$
You just forgot the constant term but you have the leading order.