In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the chain rule is a
formula that expresses the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the
composition of two
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s and in terms of the derivatives of and . More precisely, if
is the function such that
for every , 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 depends on the variable , which itself depends on the variable (that is, and are
dependent variables), then depends on as well, via the intermediate variable . 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.
Intuitive explanation
Intuitively, the chain rule states that knowing the instantaneous rate of change of relative to and that of relative to allows one to calculate the instantaneous rate of change of relative to as the product of the two rates of change.
As put by
George F. Simmons
George Finlay Simmons (March 3, 1925 – August 6, 2019) was an American mathematician who worked in topology and classical analysis. He is known as the author of widely used textbooks on university mathematics.
Life
He was born on 3 March 192 ...
: "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 , and 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.
History
The chain rule seems to have first been used by
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
. 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.
Guillaume de l'Hôpital used the chain rule implicitly in his ''
Analyse des infiniment petits''. The chain rule does not appear in any of
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
's analysis books, even though they were written over a hundred years after Leibniz's discovery.
Statement
The simplest form of the chain rule is for real-valued functions of one
real variable. It states that if ' is a function that is differentiable at a point ' (i.e. the derivative exists) and ' is a function that is differentiable at , then the composite function
is differentiable at ', and the derivative is
:
The rule is sometimes abbreviated as
:
If and , then this abbreviated form is written in
Leibniz notation as:
:
The points where the derivatives are evaluated may also be stated explicitly:
:
Carrying the same reasoning further, given ' functions
with the composite function
, if each function
is differentiable at its immediate input, then the composite function is also differentiable by the repeated application of Chain Rule, where the derivative is (in Leibniz's notation):
:
Applications
Composites of more than two functions
The chain rule can be applied to composites of more than two functions. To take the derivative of a composite of more than two functions, notice that the composite of , , and ' (in that order) is the composite of with . The chain rule states that to compute the derivative of , it is sufficient to compute the derivative of ' and the derivative of . The derivative of ' can be calculated directly, and the derivative of can be calculated by applying the chain rule again.
For concreteness, consider the function
:
This can be decomposed as the composite of three functions:
:
Their derivatives are:
:
The chain rule states that the derivative of their composite at the point is:
:
In
Leibniz's notation, this is:
:
or for short,
:
The derivative function is therefore:
:
Another way of computing this derivative is to view the composite function as the composite of and ''h''. Applying the chain rule in this manner would yield:
:
This is the same as what was computed above. This should be expected because .
Sometimes, it is necessary to differentiate an arbitrarily long composition of the form
. In this case, define
:
where
and
when
. Then the chain rule takes the form
: