Leibniz integral rule

In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the Leibniz integral rule for differentiation under the integral sign, named after <a href="/facts/Gottfried_Wilhelm_Leibniz/cTc2er83">Gottfried Wilhelm Leibniz</a>, states that for an <a href="/facts/Integral/4PeM0mJc">integral</a> of the form

∫
          
            a
            (
            x
            )
          
          
            b
            (
            x
            )
          
        
        f
        (
        x
        ,
        t
        )
        
        d
        t
        ,
      
    
    {\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}

where 
 
 
 
 −
 ∞
 <
 a
 (
 x
 )
 ,
 b
 (
 x
 )
 <
 ∞
 
 
 {\displaystyle -\infty <a(x),b(x)<\infty }
 
 and the integrands are <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> dependent on 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 the derivative of this integral is expressible as

d
                    
                      d
                      x
                    
                  
                
                
                  (
                  
                    
                      ∫
                      
                        a
                        (
                        x
                        )
                      
                      
                        b
                        (
                        x
                        )
                      
                    
                    f
                    (
                    x
                    ,
                    t
                    )
                    
                    d
                    t
                  
                  )
                
              
            
            
              
              
                
                =
                f
                
                  
                    (
                  
                
                x
                ,
                b
                (
                x
                )
                
                  
                    )
                  
                
                ⋅
                
                  
                    d
                    
                      d
                      x
                    
                  
                
                b
                (
                x
                )
                −
                f
                
                  
                    (
                  
                
                x
                ,
                a
                (
                x
                )
                
                  
                    )
                  
                
                ⋅
                
                  
                    d
                    
                      d
                      x
                    
                  
                
                a
                (
                x
                )
                +
                
                  ∫
                  
                    a
                    (
                    x
                    )
                  
                  
                    b
                    (
                    x
                    )
                  
                
                
                  
                    ∂
                    
                      ∂
                      x
                    
                  
                
                f
                (
                x
                ,
                t
                )
                
                d
                t
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)\\&=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt\end{aligned}}}

where the <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a> 
 
 
 
 
 
 
 ∂
 
 ∂
 x
 
 
 
 
 
 
 {\displaystyle {\tfrac {\partial }{\partial x}}}
 
 indicates that inside the integral, only the variation of 
 
 
 
 f
 (
 x
 ,
 t
 )
 
 
 {\displaystyle f(x,t)}
 
 with 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is considered in taking the derivative.
In the special case where the functions 
 
 
 
 a
 (
 x
 )
 
 
 {\displaystyle a(x)}
 
 and 
 
 
 
 b
 (
 x
 )
 
 
 {\displaystyle b(x)}
 
 are constants 
 
 
 
 a
 (
 x
 )
 =
 a
 
 
 {\displaystyle a(x)=a}
 
 and 
 
 
 
 b
 (
 x
 )
 =
 b
 
 
 {\displaystyle b(x)=b}
 
 with values that do not depend on 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 this simplifies to:

d
            
              d
              x
            
          
        
        
          (
          
            
              ∫
              
                a
              
              
                b
              
            
            f
            (
            x
            ,
            t
            )
            
            d
            t
          
          )
        
        =
        
          ∫
          
            a
          
          
            b
          
        
        
          
            ∂
            
              ∂
              x
            
          
        
        f
        (
        x
        ,
        t
        )
        
        d
        t
        .
      
    
    {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt.}

If 
 
 
 
 a
 (
 x
 )
 =
 a
 
 
 {\displaystyle a(x)=a}
 
 is constant and 
 
 
 
 b
 (
 x
 )
 =
 x
 
 
 {\displaystyle b(x)=x}
 
, which is another common situation (for example, in the proof of <a href="/facts/Cauchy_formula_for_repeated_integration/qwhhgTCN">Cauchy's repeated integration formula</a>), the Leibniz integral rule becomes:

d
            
              d
              x
            
          
        
        
          (
          
            
              ∫
              
                a
              
              
                x
              
            
            f
            (
            x
            ,
            t
            )
            
            d
            t
          
          )
        
        =
        f
        
          
            (
          
        
        x
        ,
        x
        
          
            )
          
        
        +
        
          ∫
          
            a
          
          
            x
          
        
        
          
            ∂
            
              ∂
              x
            
          
        
        f
        (
        x
        ,
        t
        )
        
        d
        t
        ,
      
    
    {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{x}f(x,t)\,dt\right)=f{\big (}x,x{\big )}+\int _{a}^{x}{\frac {\partial }{\partial x}}f(x,t)\,dt,}

This important result may, under certain conditions, be used to interchange the integral and partial differential <a href="/facts/Operator_(mathematics)/1VWgEdjS">operators</a>, and is particularly useful in the differentiation of <a href="/facts/Integral_transform/AB2yjHJJ">integral transforms</a>. An example of such is the <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> in <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, a variation of the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a>, which can be differentiated to generate the <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a>. Whether Leibniz's integral rule applies is essentially a question about the interchange of <a href="/facts/Limit_(mathematics)/j8JBhj4e">limits</a>.

Leibniz integral rule open-in-new

Leibniz integral rule