Derivative under Integral

Obviously this has become lot popular after Richard Feynnman's book. Surely you must be joking where he makes a reference about this method. The method also goes by the name of Leibnitz's method and can be found with proof in the Schaum's Advanced calculus. Wikipedia also has some nice worked out examples under the above heading. Its fairly simple to explain. Suppose your integral is a function of some variable 'a' for example
f(a) = 1/(1-a*cos(x))
int(f(a),x=u1(a)..u2(a)) note that u1(a) and u2(a) are also function of a
Then the derivative of the above function (integral) of a w.r.t a is
int(del(f(a)/da,x = u1(a)..u2(a))+del(u2(a))/da*f(u2(a))-del(u1(a))/da*f(u1(a))
With this we can prove lots of neat results for example there are situations when the integral is shown to be equal to some result and can be verified by taking derivative on both sides.