Fisher Information Matrix is defined as
\[\begin{equation} [\mathcal{I}(\theta)]_{i, j}\equiv\mathrm{E}\left[\left(\frac{\partial}{\partial \theta_{i}} \log f(X ; \theta)\right)\left(\frac{\partial}{\partial \theta_{j}} \log f(X ; \theta)\right) \right] \tag{1} \end{equation}\]
Under regularity conditions,
\[\begin{equation} \begin{aligned} \mathrm{E}\left[\frac{1}{f(X ; \theta)} \frac{\partial^{2}}{\partial \theta_{i} \theta_{j}} f(X ; \theta)\right] &= \int_{x} \frac{1}{f(X ; \theta)} \frac{\partial^{2} f(X ; \theta)}{\partial \theta_{i} \partial \theta_{j}} f(X ; \theta)dx \\ &= \frac{\partial^2}{\partial \theta_{i} \partial \theta_{j}} \int_{x} f(X ; \theta) dx \\ &= \frac{\partial^{2} 1}{\partial \theta_{i} \partial \theta_{j}} \\ &= 0 \end{aligned} \tag{2} \end{equation}\]
\[\begin{equation} \begin{aligned} -\mathrm{E}\left[\frac{\partial^{2}}{\partial \theta_{i} \partial \theta_{j}} \log f(X ; \theta) \right] &= -\mathrm{E}\left[ \frac{\partial \left(\frac{1}{f(X ; \theta)} \frac{\partial}{\partial \theta_{i}} f(X ; \theta)\right)} {\partial \theta_j} \right] \\ &=-\mathrm{E} \left[ \frac{-1}{f^2(X ; \theta)}\frac{\partial}{\partial \theta_{i}} f(X ; \theta) \frac{\partial}{\partial \theta_{j}} f(X ; \theta) + \frac{1}{f(X ; \theta)} \frac{\partial^2}{\partial \theta_{i}\theta_{j}} f(X ; \theta) \right]\\ &= \mathrm{E}\left[\left(\frac{\partial}{\partial \theta_{i}} \log f(X ; \theta)\right)\left(\frac{\partial}{\partial \theta_{j}} \log f(X ; \theta)\right)\right] - \mathrm{E}\left[ \frac{1}{f(X ; \theta)} \frac{\partial^{2} f(X ; \theta)}{\partial \theta_{i} \partial \theta_{j}} \right] \\ &= \mathrm{E}\left[\left(\frac{\partial}{\partial \theta_{i}} \log f(X ; \theta)\right)\left(\frac{\partial}{\partial \theta_{j}} \log f(X ; \theta)\right)\right] - 0\\ &= [\mathcal{I}(\theta)]_{i, j} \end{aligned} \tag{3} \end{equation}\]
Overall, under regularity conditions, we have \[\begin{equation} [\mathcal{I}(\theta)]_{i, j} \equiv \mathrm{E}\left[\left(\frac{\partial}{\partial \theta_{i}} \log f(X ; \theta)\right)\left(\frac{\partial}{\partial \theta_{j}} \log f(X ; \theta)\right)\right] = -\mathrm{E}\left[\frac{\partial^{2}}{\partial \theta_{i} \partial \theta_{j}} \log f(X ; \theta)\right] \tag{4} \end{equation}\]