Physical motivation for differentiation under the integral I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s f(x,s)\,dx$$ given some regularity assumptions. I was trying to think of some relevant physical interpretations of this. 
One that I camp up with (that I think is rather weak) is: the total force exerted by the walls of a chamber holding a gas is defined by an integral. We might want to ask how that function is changing with respect to some parameter of the gas, so we'd differentiate under the integral.
Anyone have a better one?
 A: I guess a simple example would be the derivation of the continuity equation.
It will first be useful to define the phase space probability density, denoted by $f=f(q_1,\dots,q_{N},p_1,\dots,p_{N},t) \equiv f(\mathbf{x},t)$, that tells us the probability of finding a system near $(q_1,\dots,q_{N},p_1,\dots,p_{N})$ at a time $t$. Therefore:
\begin{equation}
\mathrm{d}N = f(\mathbf{x},t) \; \mathrm{d}x
\end{equation}
relates phase space volume elements to the number of particles $N$ in that volume. In other words, $f$ is the number of volume elements occupying the differential phase space volume $\mathrm{d}x$ and:
\begin{equation}
N = \int \mathrm{d}x \; f(\mathbf{x},t) 
\end{equation}
gives the number of particles in the ensemble.
Now, remember that the flux through a surface is given by:
\begin{equation}
j = \int f \mathbf{v} \cdot \mathrm{d} \mathbf{S}
\end{equation}
where $\mathbf{v}$ denotes the flow velocity of the particles. Subsequently, using the $N$-dimensional divergence theorem, the total flux out of a volume $V$ bounded by $\partial V$ is given by:
\begin{equation}
\oint_{\partial V} f \mathbf{v} \cdot \mathrm{d} \mathbf{S} = \int_V  \mathbf{\nabla}_{\mathbf{x}} \cdot \left( f \mathbf{v} \right) \; \mathrm{d}x
\end{equation}
where $\mathbf{\nabla}_{\mathbf{x}}$ is the $2N$ dimensional gradient on the phase space:
\begin{equation}
\mathbf{\nabla}_{\mathbf{x}} = \left(\frac{\partial}{\partial  q_1}, \dots , \frac{\partial}{\partial  q_{N}},\frac{\partial}{\partial  p_1}, \dots , \frac{\partial}{\partial  p_{N}} \right)
\end{equation}
On the other hand, using the rule of differentiation under the integral sign, we can write the rate of decrease of particles in the volume as:
\begin{equation}
\frac{d N}{d t} = - \frac{d}{d t} \int_V \mathrm{d}x \; f = - \int_V \mathrm{d}x \; \frac{\partial f}{\partial t}
\end{equation}
Therefore, demanding equality between the fourth equation and the sixth equation gives (because the flux out of the volume must equal the reate of decrease of particles in the volume):
\begin{equation}
\int_V  \mathbf{\nabla}_{\mathbf{x}} \cdot \left( f \mathbf{v} \right)  \; \mathrm{d}x = - \int_V \mathrm{d}x \; \frac{\partial f}{\partial t}
\end{equation}
which is the continuity equation.
We also need to use the "differentiation under the integral sign" if we want to derive probability current in quantum mechanics. However, I have not taken this example, because I do not know your background and if/how much knowledge of QM you have. Also for Ehrenfest theorem in QM we need to use this rule.
Another (more mathematical) example when this is useful is for contour integration (and in particular for Cauchy's integral formula).
A: For example, consider some water flow in the space, in which the density $\rho(x,t)$ fluctuates in space and in time. You might be interested in how the mass inside some fixed volume $V$ changes over time. The mass is equal to
$$
M(t)=\int_V{\rho(x,t)\mathrm{d}x},
$$
therefore the "mass flow rate", using the rule you mentioned, is equal to
$$
\frac{\mathrm{d}}{\mathrm{d}t}M(t)=\int_V \partial_t\rho(x,t)\mathrm{d}x.
$$
Other examples include energy, probability, momentum or state densities instead of mass density.
A: Possibly the most ubiquitous use is to calculate the derivative of a functional which is defined as an integral, e.g., of the Lagrangian along a path.  Typical application---computing the equations of motion as the euler equations of a problem of the calculus of variations
