What is it?

Differentiating $\int_{a(x)}^{b(x)} f(x,t) dt$ with respect to $x$, you get:

$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt$

Example

In the interdependent value auction setting, we have the bidder whose true type is $x$ but choose a value $x'$ to maximise the following expected payoff function:

$\int_{0}^{x'} v(x,x_j) f_{X_j|X_i}(x_j | X_i = x) dx_j - F_{X_j|X_i}(x' | X_i = x)m(x, x')$

Where $m(x,x')$ denotes the payment bidder $i$ has to make when he/she wins.

If we assume the strategy $\beta(x)$ to be the equilibrium strategy, then revealing the truth, i.e. choosing $x' = x$, must yield the highest expected payoff.

Therefore it must also satisfy the necessary condition for maximum when evaluating the FOC at $x' = x$:

$\frac{d}{dx'} \int_{0}^{x'} v(x,x_j) f_{X_j|X_i}(x_j | X_i = x) dx_j = v(x, x') f_{X_j|X_i}(x' | X_i = x)$

Thus we have:

$v(x,x') f_{X_j|X_i}(x' | X_i = x)_{|x' = x} = \frac{d}{dx'} F_{X_j|X_i}(x' | X_i = x)m(x,x')_{|x' = x}$

i.e. $v(x,x) f_{X_j|X_i}(x | X_i = x) = \frac{d}{dx'} F_{X_j|X_i}(x | X_i = x)m(x,x)$

Useful link

http://www.econ.yale.edu/~pah29/409web/leibniz.pdf