Leibniz Integral Rule

Then the derivative of I with respect to its upper limit is given by 

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Applying Leibniz s rule for the dififerentiation of a parameter integral [76], it follows from Eqs. (89) and (95) that... 

This integral equation can be transformed into an ordinary differential equation by taking its derivative relative to t and applying Leibniz s rule to the integral... 

The trouble is now that the source term does not include the sum of sines, so we will use a trick resting on the Leibniz s rule for differentiating integrals. A particular solution of the diffusion equation with radiogenic accumulation is... 

Applying the Leibniz s rule to the integral on the right-hand side, we get... 

We could take the derivative of this expression and apply Leibniz s rule, but we know that exponentials have the property that their integrals and derivatives are linearly related. We therefore try the solution... 

Hence, we may again apply Leibniz s rule for the differentiation of a paiain-eter integral to obtain... 

Baumeister and Hamill [32] solved the hyperbolic heat conduction equation in a semi-infinite medium subjected to a step change in temperature at one of its ends using the method of Laplace transform. The space-integrated expression for the temperature in the Laplace domain had the inversion readily available within the tables. This expression was differentiated using Leibniz s rule, and the resulting temperature distribution was given for x > X as... 

The Leibniz rule (see Integral Calculus ) can be used to show the equivalence of the initial-value problem consisting of the second-order differential equation d2y/cbd + A(x)(dy/dx) + B(x)y = fix) together with the prescribed initial conditions y(a) = y . y (a) = if, to the integral equation. 

Hint recall the Leibniz rule for differentiating an integral in which the limits of integration are variable. 

We can also show that the latent heat is simply related to the slope of the vapor pressure curve P (T). Let us differentiate the vapor pressure in (8.2.21) wrt temperature we recognize that the integration limits and v must change with temperature, so we apply the Leibniz rule for differentiating such integrals (Appendix A). The result is... 

This is the Leibniz rule for differentiating definite integrals. In those special cases in which one or both limits (a and b) are constants, independent of x, then (A.11.7) simplifies accordingly. 

Differentiation with respect to t (the rule of Leibniz is applied to the integral) leads to... 

Equations 5.17 and 5.18 are formally equivalent to those we used for the unwrapped model of the screw extruder, and they provide the starting point for considering cases with a channel depth that varies with x. It is often useful to employ a different formulation when H varies with both x and z, however, for in that case we cannot expect either qx or Qz to remain constant. Now, we recall the Leibniz rule for differentiating an integral ... 

For all fractional operators, the rules of classical derivatives and integrals (linearity, semigroup, Leibniz rule, etc.) hold. 

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