  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 $$D^If(\eta)(x)= D^{L}f(\eta)(x)=
\lim_{s\downarrow 0} \frac 1 s \nabla f(\eta+s \delta_\cdot)(x) = \nabla
\big\{D^E f (\eta)\big\}(x), \ \ (x,\eta)\in M\times\mathbb M,$$ where $\nabla$
is the gradient on $M$, $\delta_x$ is the Dirac measure at $x$, and
$$D^Ef(\eta)(x):= \lim\limits_{s\downarrow 0} \frac { f(\eta+s
\delta_x)-f(\eta)} s,\ \ x\in M$$ is the extrinsic derivative of $f$ at
$\eta\in \mathbb M$. 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.
