Leibnitz's Rule Statement And It's Proof

        WELCOME TO MATHEMATICS
      

     In this mathematics session I shall prove that , under suitable conditions, ' the derivative of the integral and the integral of the derivative are equal' , and consequently , 'the two repeated integrals are equal for continuous functions'.
        

Leibnitz's Rule In Mathematics:

               If f is defined and continuous on the rectangle R = [a,b;c,d] , and if 

  (i)  fₓ(x,y) exists and is continuous on the rectangle R , and 

                    d

  (ii) g(x) = ∫ f(x,y) dy , for x∈ [a,b]

                    c

then g is differentiable on  [a,b]and 

                          d

             g'(x) = ∫ fₓ(x,y) dy 

                         c

                       d                    d

i.e.,      d/dx {∫ f(x,y) dy }=∫ ∂f(x,y)/∂x dy

                      c                    c

 Proof Of Leibnitz's Rule In Mathematics :

           Since fₓ (∂f/∂x) exists on R , therefore,  for each y∈ [c,d] , and each h≠ 0 , it follows by the Lagrange's Mean Value Theorem , that 

      f(x+h , y) - f(x,y) = h fₓ(x+θh , y),

                                      for some 0<θ<1

         Now fₓ being continuous is integrable on [c,d] for each x∈ [a,b] , therefore g(x) is well defined function on [a,b].

                                          d

    g(x+h) - g(x) /h = 1/h ∫ {f(x+h ,y) - f(x,y)}dy

                                         c

                                 d

                             = ∫ fₓ(x+θh ,y) dy , 0<θ<1

                                c                   .......(1)

    Let ε>0 be given . Then by continuity and uniform continuity of fₓ on R , ∃ δ>0 , such that if (x,y) ,(x',y') ∈ R with 

        |x - x'| <δ , |y - y'| <δ , then 

          |fₓ(x,y) - fₓ(x',y')| <ε/(d-c)   .........(2)

From equations (1) and (2) , we obtain 

                                      d

   | g(x+h) - g(x) / h - ∫ fₓ(x,y) dy |

                                     c

                       d

                   ≤ ∫ |fₓ(x+θh , y) - fₓ(x,y)| dy

                      c

                   <  ε(d-c)/ (d-c) = ε , 0<|h|<δ
                                                         d
Here g'(x) = lim g(x+h) - g(x) /h ∫fₓ(x,y) dy
                     h-->0                          c

                                                   (Proved)


To understand this rule we have to solve an example , so let us take an example .

Example Of Leibnitz's Rule In Mathematics :

        Show that 

         π/2

         ∫ log(1-x²sin²θ) dθ 

         0

               = π log(1+√(1-x²) - π log 2 , if |x|<1

Proof :

       The function log (1-x² sin²θ) is well defined in the rectangle [-1,1;0,π/2] and satisfies the conditions of the Leibnitz's  rule .                  π/2

      Let     g(x) = ∫ log (1-x² sin²θ) dθ , |x|<1

                           0                      ..........(1)

     By differentiating under the integral sign , w.r.t. x , we get 

             π/2

  g'(x) = ∫ -2x sin² θ dθ/(1-x² sin²θ) 

             0

       

                   π/2

         = 2/x ∫ (1-x²sin²θ -1) dθ /(1-x²sin²θ) dθ

                  0                                        x≠ 0

                         π/2

        = π/x -2/x ∫ dθ/(1-x² sin²θ) , put cot θ = 1

                         0

                          ∞

       = π/x - 2/x ∫ dt /(1+t²-x²)

                         0

                                                              ∞

       = π/x - 2/x√(1-x²) tan⁻¹ t/√(1-x²) ]

                                                              0

       = π/x - π/x√(1-x²)

Integrating w.r.t. x , we obtain 

    g(x) = π log x - π log {1-√(1-x² ) / x} + c

                   where c is an arbitrary constant 

           = π log {x²/(1-√(1-x² )} + c

           = π log {x²(1+√(1-x² ) / 1-(1-x²)} + c

           =  π log (1+√(1-x²)) + c 

But g(0) = 0 , by equation(1) , therefore 

                         c = -π log 2

Hence , g(x) = π log (1+√(1-x² ) - π log 2,

                                for     |x| < 1 

                                                 (Proved)

About The Scientists:

        Gottfried Wilhelm Leibniz was a prominent German polymath and one of the most important logicians, mathematicians and natural philosophers of the Enlightenment.Readmore.