Reparametrization of multivariable fisher information

The task is to prove that reparametrization of multivariable Fisher information with diffeomorphism $\eta: \Theta \rightarrow H$ is $$I(\eta)=\bigg(\frac{\partial \theta^T}{\partial \eta}\bigg)I(\theta) \bigg(\frac{\partial \theta}{\partial \eta^T}\bigg),$$ for given $I(\theta)$.

My solution:

We know that multivariable Fisher information is $$I(\theta)=\mathbb{E}\bigg\{\frac{\partial}{\partial\theta}\log{(f_\theta(x))}\bigg[\frac{\partial}{\partial \theta}\log{(f_\theta(x))}\bigg]^T\bigg\}.$$

We can apply chain rule $$\frac{\partial}{\partial \eta}= \frac{\partial \theta}{\partial \eta}\frac{\partial}{\partial \theta} \Rightarrow \frac{\partial}{\partial \theta} = \bigg(\frac{\partial \theta}{\partial \eta}\bigg)^{-1}\frac{\partial}{\partial \eta} %\theta'(\eta)\frac{\partial}{\partial\theta}$$

Applying it to the first equation and changing $\theta \mapsto \eta$ we obtain $$I(\eta)=\mathbb{E}\bigg\{\frac{\partial\theta}{\partial\eta}\frac{\partial}{\partial\theta}\log{(f_\eta(x))}\bigg[\frac{\partial\theta}{\partial\eta}\frac{\partial}{\partial \theta}\log{(f_\eta(x))}\bigg]^T\bigg\}.$$ Now it's somewhere close to solution, what I would like to do is to take this $\big(\frac{\partial\theta}{\partial\eta}\big)$s out of the expected value and tell that it is done. But I don't think it is correct.

Can anyone help?