We characterise the link of derivatives in measure, which are introduced in [AKR,Card,ORS] respectively by different means, for functions on the space $\mathbb M$ of finite measures over a Riemannian manifold $M$. For a reasonable class of functions $f$, the extrinsic derivative $D^Ef$ coincides with the linear functional derivative $D^Ff$, the intrinsic derivative $D^If$ equals to the $L$-derivative $D^Lf$, and $$D^If(η)(x)= D^{L}f(η)(x)= \lim_{s\downarrow 0} \frac 1 s \nabla f(η+s δ_\cdot)(x) = \nabla \big\{D^E f (η)\big\}(x), \ \ (x,η)\in M\times\mathbb M,$$ where $\nabla$ is the gradient on $M$, $δ_x$ is the Dirac measure at $x$, and $$D^Ef(η)(x):= \lim\limits_{s\downarrow 0} \frac { f(η+s δ_x)-f(η)} s,\ \ x\in M$$ is the extrinsic derivative of $f$ at $η\in \mathbb M$. This gives a simple way to calculate the intrinsic or $L$-derivative, and is extended to functions of probability measures. %This provides a simple way to calculate the intrinsic/Lions derivative.