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Concept
Chain rule
Summary
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation,
or, equivalently,
The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y. In this case, the chain rule is expressed as
and
for indicating at which points the derivatives have to be evaluated.
In integration, the counterpart to the chain rule is the substitution rule.
Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change.
As put by George F. Simmons: "If a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man."
The relationship between this example and the chain rule is as follows. Let z, y and x be the (variable) positions of the car, the bicycle, and the walking man, respectively. The rate of change of relative positions of the car and the bicycle is Similarly, So, the rate of change of the relative positions of the car and the walking man is
The rate of change of positions is the ratio of the speeds, and the speed is the derivative of the position with respect to the time; that is,
or, equivalently,
which is also an application of the chain rule.
The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of as the composite of the square root function and the function . He first mentioned it in a 1676 memoir (with a sign error in the calculation). The common notation of the chain rule is due to Leibniz.
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