Derivative of the Wasserstein Metric between two Gaussians

Question

I am trying to take the derivative of the squared Wasserstein metric between two Gaussian probability densities, which is given by $W_2^2(q_0, q_1) = \| \boldsymbol{\mu}_0 - \boldsymbol{\mu}_1 \|_2^2 + \text{tr}[\boldsymbol{\Sigma}_0 + \boldsymbol{\Sigma}_1 - 2 (\boldsymbol{\Sigma}_0^{1/2} \boldsymbol{\Sigma}_1 \boldsymbol{\Sigma}_0^{1/2})^{1/2} ]$, where $q_0 = \mathcal{N}(\boldsymbol{\mu}_0,\boldsymbol{\Sigma}_0)$ and $q_1 = \mathcal{N}(\boldsymbol{\mu}_1,\boldsymbol{\Sigma}_1)$, are Gaussian densities. I need the derivative $\frac{\partial W_2^2(q_0,q_1)}{\partial \boldsymbol{\Sigma}_1}$; however, I am unsure how to take the derivative of the matrix square-root within the trace term.