User:Support.and.Defend/chain rule
From Wikipedia, the free encyclopedia
In calculus, the chain rule is a method to compute the derivative of the composition of two or more functions.
If a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, that is y = y(u(x)) , then the rate of change of y with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x. Schematically,
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another. For example, if f is a function of g which is in turn a function of h, which is in turn a function of x, that is
,
the derivative of f with respect to x is given by
and so on.
The chain rule has broad applications in Physics, Chemistry, and Engineering, as well as being used to study related rates in many disciplines. For example, if a skydiver is falling at 60 miles per hour and the barometric pressure increases at 3 psi per mile fallen, then the skydiver experiences an increasing pressure at a rate of 60 mi/hr * 3 psi/mi = 180 psi per hour. The chain rule can also be generalized to multiple variables in cases where the nested functions depend on more than one variable.