Trapezoidal rule that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as "Monkhorst-Pack integration". For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule. The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has
General Leibniz rule General Leibniz rule In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if formula_1 and formula_2 are formula_3-times differentiable functions, then the product formula_4 is also formula_3-times differentiable and its formula_3th derivative is given by where formula_8 is the binomial coefficient and formula_9 This can be proved by using the product rule and mathematical induction. If, for example, , the rule gives an expression for the second derivative of a product of two functions: The formula can be generalized to the product of "m"