The chain rule is a method to compute the derivative of the functional composition of two or more functions.
If a function
depends on a variable
, which in turn depends on another variable
, that is
, then the rate of change of
with respect to
can be computed as the rate of change of
with respect to
multiplied by the rate of change of
with respect to
.
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another. [1] For example, if
is a function of
which is in turn a function of
, which is in turn a function of
, that is

the derivative of
with respect to
is given by
and so on.
A useful mnemonic is to think of the differentials as individual entities that can be canceled algebraically, such as

However, keep in mind that this trick comes about through a clever choice of notation rather than through actual algebraic cancellation.
The chain rule has broad applications in physics, chemistry, and engineering, as well as being used to study related rates in many disciplines. The chain rule can also be generalized to multiple variables in cases where the nested functions depend on more than one variable.
Examples[edit]
Example I[edit]
Suppose that a mountain climber ascends at a rate of
. The temperature is lower at higher elevations; suppose the rate by which it decreases is
per kilometer. To calculate the decrease in air temperature per hour that the climber experiences, one multiplies
by
, to obtain
. This calculation is a typical chain rule application.
Example II[edit]
Consider the function
. It follows from the chain rule that
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Function to differentiate
|
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Define as inside function
|
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Express in terms of
|
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Express chain rule applicable here
|
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Substitute in and
|
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Compute derivatives with power rule
|
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Substitute back in terms of
|
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Simplify.
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Example III[edit]
In order to differentiate the trigonometric function

one can write:
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Function to differentiate
|
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Define as inside function
|
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Express in terms of
|
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Express chain rule applicable here
|
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Substitute in and
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Evaluate derivatives
|
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Substitute in terms of .
|
Example IV: absolute value[edit]
The chain rule can be used to differentiate
, the absolute value function:
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Function to differentiate
|
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Equivalent function
|
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Define as inside function
|
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Express in terms of
|
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Express chain rule applicable here
|
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Substitute in and
|
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Compute derivatives with power rule
|
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Substitute back in terms of
|
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Simplify
|
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Express as absolute value.
|
Example V: three nested functions[edit]
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
, sequential application of the chain rule yields the derivative as follows (we make use of the fact that
, which will be proved in a later section):
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Original (outermost) function
|
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Define as innermost function
|
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as middle function
|
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Express chain rule applicable here
|
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Differentiate f(g)[2]
|
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Differentiate
|
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Differentiate
|
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Substitute into chain rule.
|
Chain Rule in Physics[edit]
Because one physical quantity often depends on another, which, in turn depends on others, the chain rule has broad applications in physics. This section presents examples of the chain rule in kinematics and simple harmonic motion. The chain rule is also useful in electromagnetic induction.
Physics Example I: relative kinematics of two vehicles[edit]
One vehicle is headed north and currently located at

; the other vehicle is headed west and currently located at

. The chain rule can be used to find whether they are getting closer or further apart.
For example, one can consider the kinematics problem where one vehicle is heading west toward an intersection at 80mph while another is heading north away from the intersection at 60mph. One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the northbound vehicle is 3 miles north of the intersection and the westbound vehicle is 4 miles east of the intersection.
Big idea: use chain rule to compute rate of change of distance between two vehicles.
- Plan
- Choose coordinate system
- Identify variables
- Draw picture
- Big idea: use chain rule to compute rate of change of distance between two vehicles
- Express
in terms of
and
via Pythagorean theorem
- Express
using chain rule in terms of
and 
- Substitute in

- Simplify.
Choose coordinate system:
Let the
-axis point north and the x-axis point east.
Identify variables:
Define
to be the distance of the vehicle heading north from the origin and
to be the distance of the vehicle heading west from the origin.
Express
in terms of
and
via Pythagorean theorem:

Express
using chain rule in terms of
and
:
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Apply derivative operator to entire function
|
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Sum of squares is inside function
|
![{\displaystyle ={\frac {(x^{2}+y^{2})^{-{\frac {1}{2}}}}{2}}\left[{\frac {d}{dt}}(x^{2})+{\frac {d}{dt}}(y^{2})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b656c37ba8c39165514d9c4f0aa65607823dec) |
Distribute differentiation operator
|
![{\displaystyle ={\frac {(x^{2}+y^{2})^{-{\frac {1}{2}}}}{2}}\left[2x\cdot {\frac {dx}{dt}}+2y\cdot {\frac {dy}{dt}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb61fdaf2803d1535e1cc73516ecb68202d807b) |
Apply chain rule to and
|
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Simplify.
|
Substitute in
and simplify
|
|
|
|
|
|
|
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Consequently, the two vehicles are getting closer together at a rate of
.
Physics Example II: harmonic oscillator[edit]
An undamped spring-mass system is a simple harmonic oscillator.
If the displacement of a simple harmonic oscillator from equilibrium is given by
, and it is released from its maximum displacement
at time
, then the position at later times is given by

where
is the angular frequency and
is the period of oscillation. The velocity,
, being the first time derivative of the position can be computed with the chain rule:
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Definition of velocity in one dimension
|
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Substitute
|
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Bring constant outside of derivative
|
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Differentiate outside function (cosine)
|
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Bring negative sign in front
|
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Evaluate remaining derivative
|
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Simplify.
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The acceleration is then the second time derivative of position, or simply
.
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Definition of acceleration in one dimension
|
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Substitute
|
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Bring constant term outside of derivative
|
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Differentiate outside function (sine)
|
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Evaluate remaining derivative
|
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Simplify.
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From Newton's second law,
, where
is the net force and
is the object's mass.
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Newton's second law
|
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Substitute
|
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Simplify
|
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Substitute original .
|
Thus it can be seen that these results are consistent with the observation that the force on a simple harmonic oscillator is a negative constant times the displacement.
Chain Rule in Chemistry[edit]
The chain rule has many applications in Chemistry because many equations in Chemistry describe how one physical quantity depends on another, which in turn depends on another. For example, the ideal gas law describes the relationship between pressure, volume, temperature, and number of moles, all of which can also depend on time.
Chemistry Example I: Ideal Gas Law[edit]
Isotherms of an ideal gas. The curved lines represent the relationship between pressure and volume for an ideal gas at different temperatures: lines which are further away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram) represent higher temperatures.
Suppose a sample of
moles of an ideal gas is held in an isothermal (constant temperature,
) chamber with initial volume
. The ideal gas is compressed by a piston so that its volume changes at a constant rate so that
, where
is the time. The chain rule can be employed to find the time rate of change of the pressure.[3] The ideal gas law can be solved for the pressure,
to give:

where
and
have been written as explicit functions of time and the other symbols are constant. Differentiating both sides yields

where the constant terms
have been moved to the left of the derivative operator. Applying the chain rule gives

where the power rule has been used to differentiate
, Since
,
. Substituting in for
and
yields
.

Chemistry Example II: Kinetic Theory of Gases[edit]
The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.
A second application of the chain rule in Chemistry is finding the rate of change of the average molecular speed,
, in an ideal gas as the absolute temperature
, increases at a constant rate so that
, where
is the initial temperature and
is the time.[3] The kinetic theory of gases relates the root mean square of the molecular speed to the temperature, so that if
and
are functions of time,

where
is the ideal gas constant, and
is the molecular weight.
Differentiating both sides with respect to time yields:

Using the chain rule to express the right side in terms of the with respect to temperature,
, and time,
, respectively gives

Evaluating the derivative with respect to temperature,
, yields

Evaluating the remaining derivative with respect to
, taking the reciprocal of the negative power, and substituting
, produces

Evaluating the derivative with respect to
yields

which simplifies to

Proof of the chain rule[edit]
Suppose
is a function of
which is a function of
(it is assumed that
is differentiable at
and
, and
is differentiable at
.
To prove the chain rule we use the definition of the derivative.

We now multiply
by
and perform some algebraic manipulation.

Note that as
approaches
,
also approaches
. So taking the limit as of a function as
approaches
is the same as taking its limit as
approaches
. Thus

So we have

Exercises[edit]
1. Evaluate

if

, first by expanding and differentiating directly, and then by applying the chain rule on

where

. Compare answers.


2. Evaluate the derivative of

using the chain rule by letting

and

.


Solutions
References[edit]
External links[edit]