The product rule of differentiation will be used when prove the Rao-Cramer lower bound.
\[\frac{d(uvw)}{dx}=\frac{du}{dv}vw+u\frac{dv}{dx}w+uv\frac{dw}{dx}\]
\[\frac{d}{dx}\left[\prod_{i=1}^kf_i(x) \right]=\sum_{i=1}^k\left[(\frac{d}{dx}f_i(x))\prod_{j=1,j \neq i}^k f_j(x)\right]=(\prod_{i=1}^kf_i(x))(\sum_{i=1}^k\frac{f_i'(x)}{f_i(x)})\]