yes there is a lebnitz rule of differentiation and u can easily find the topic in the arihant book for differentiation or u can search the internet

Leibnitz theorem is for successive differentiation.If fⁿ(x) denotes the nth derivative of f(x) then the nth derivative of composite fun. f(x).g(x) is

d/dxⁿ[f(x).g(x)]=nc_of(x)gⁿ(x)+nc₁f'(x)g^n-1(x)+.......+nc_nfⁿ(x)g(x)

 i think there is one more formula.......

F(x)= lim ( g(x) to h(x) ) integration of u(x)

when we diffrentiate this according to newton lebnitz formula ....we get

F'(x) = u(g(x)) * g'(x) - u(h(x)) * h'(x)

the very correct lebnitz theorem is as follows:-                                                                                                            

i m using {"} for integral

d/dx  [ {"}_A(x)^B(x)  f(x,t)dt  ]    =    [ {"}_A(x)^B(x)  d'/d'x { f(x,t) dt } ]     +  dB/dx f(x,B)   -  dA/dx f(x,A) 

                                         here  d'/d'x  denotes partial differentiation wrt variable 't'

                                                    A(x) n B(x) denote  functions of 'x'

IF UR GIVEN FUNCTION IS JUST IN ONE VARIABLE THEN THE TERM WITH PARTIAL DIFFERENTIATION BECOMES ZERO        ........................... 

                                            AND   THE  RESULT  IS  WHAT  GIVEN BY BIGGHE  

************* SOMETIMES IT IS IMMPOSSIBLE TO INTEGRATE A GIVEN FUNCTION WITH THE GIVEN VARIABLE. IN THAT CASE USING THIS METHOD BECOMES EXTREMELY USEFUL AND NECESSARY       i.e. first                        differentiate  wrt one variable and integrate wrt other********************************       and then use standard integration   ! ! ! ! !