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
Chain rule a variable "z" depends on the variable "y", which itself depends on the variable "x", so that "y" and "z" are therefore dependent variables, then "z", via the intermediate variable of "y", depends on "x" as well. The chain rule then states, The two versions of the chain rule are related; if formula_4 and formula_5, then In integration, the counterpart to the chain rule is the substitution rule. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of formula_7 as the composite of the square root function and the