Fisher information for sample $x$ in experiment $(\Omega, \mathcal{F}, P_\theta)$ is defined as $$Var \left[\nabla_{\theta}\ell(\theta, x) \right] = \mathbb{E}\left[[\nabla_{\theta} \ell(\theta, x)] [\nabla_{\theta}\ell(\theta, x)]^T\right] $$ where $\ell(\theta, x) = \log(f(x|\theta)$.
I do not understand how this definition is applied to a very basic and well known example: Let $x \sim U(0,\theta)$. In this case the probability density of $x$ is $$ f(x_i|\theta) = \begin{cases} \frac{1}{\theta} & x\in [0, \theta]\\ 0 & \text{otherwise} \end{cases} $$
It looks to me that the density it is not differentiable with respect to $\theta$ at $\theta = x$ , consequently $\nabla_{\theta}\ell(\theta, x)$ is not defined for $\theta = x$.
I know this is a very basic example, but for some reason I couldn't find a full derivation of fisher information for this case, all I see is the stated result that it is $\frac{1}{\theta^2}$.
I would appreciate anyone pointing out to what do I miss here... Thanks!
