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I was reading up on the Stochastic Value Gradients paper by Heess et al. In the paper, they describe a recursive process to calculate path-wise derivatives via equations (3) and (4), at the bottom of page 2. Slightly above these equations, they use the notation $g_x = \frac{\partial g}{\partial x}$, which I assume is being used in both equations (3) and (4). However, when they expand on the derivation of equation (4), in the appendix of the paper, they use total gradients, which has confused me.

Can anyone clear up what $V_\theta$ and $V_s$ actually represent in equations (3) and (4), are the partial or total derivatives?

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