The process is F'(t)=∫? g(x，t)/？ t dx

=∫ 2t/(x？ +t？ )dx

=2arctan(x/t)| [0，2t+ 1]

=2 arctangent [(2t+ 1)/t]

So f' (-1) = 2 arctan1= π/2.

It cannot be solved by the method of variable upper bound integral derivative, because the integrand function is a function of t.

Modern definition of function

Given a number set A, assuming that the element in it is X, the corresponding rule F is applied to the element X in A, denoted as f(x), and another number set B is obtained. Assume that the element in B is Y, and the equivalence relation between Y and X can be expressed as y=f(x). The concept of function includes three elements: domain A, range B and corresponding rule F, among which the core is corresponding rule F, which is the essential feature of function relationship.