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.