This question is in regards to the Deepmind paper on DDPG: https://arxiv.org/pdf/1509.02971v5.pdf, specifically regarding Equation 6.

Most (all?) implementations of the DDPG algorithm that I've seen compute the gradient update to the actor network by $\nabla(J)=\nabla_{\mu(s|\theta)}(Q(s,\mu(s|\theta))\nabla_{\theta}(\mu(s|\theta))$, where $\theta$ represents the actor network's parameters, $\mu$ represents the actor network, $Q$ repesents the critic network, and $s$ represents the state input. This formula, as is shown in the paper, is derived by taking the chain rule of $\nabla(J)=\nabla_{\theta}(Q)$.

My question is why can I not just use this second equation--why do all implementations seem to explicitly compute gradients of $Q$ wrt $\mu(s)$ and multiply them by the gradient of $\mu(s)$ wrt $\theta$? Is it incorrect to just use the equation before applying chain rule as the update? They seem to be identical, and with auto-grad software packages it would seem to be easier to use the simpler equation.