Leibniz s rule

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

Given /(x, t) a function of time and of another variable x, and given the time-dependent limits a(t) and P(t), Leibniz s rule states 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... 

Attention must be paid to the sign in deriving the function ierfc, since using Leibniz s rule... 

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... 

The Fukui function, f(r), measures the response of a reactant to changes in both N and v(r). That the Fukui function is normalized to unity follows from Leibniz s rule ... 

Finally, we differentiate Kij as indicated here, using Leibniz s rule to evaluate the multiple derivatives ... 

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... 

Recall that R is itself a function of z. To obtain the rate of change, d

product rule for differentiation and Leibniz s rule for differentiating... 

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