Calculus/Chain Rule

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Chain Rule

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 .

Chain Rule

If a function is composed to two differentiable functions and , so that , then is differentiable and,

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 | edit source]

Example I[edit | edit source]

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 | edit source]

Consider the function . It follows from the chain rule that

Function to differentiate
Define as inside function
Express in terms of
Express chain rule applicable here
Substitute in and
Compute derivatives with power rule
Substitute back in terms of
Simplify.

Example III[edit | edit source]

In order to differentiate the trigonometric function

one can write:

Function to differentiate
Define as inside function
Express in terms of
Express chain rule applicable here
Substitute in and
Evaluate derivatives
Substitute in terms of .

Example IV: absolute value[edit | edit source]

The chain rule can be used to differentiate , the absolute value function:

Function to differentiate
Equivalent function
Define as inside function
Express in terms of
Express chain rule applicable here
Substitute in and
Compute derivatives with power rule
Substitute back in terms of
Simplify
Express as absolute value.

Example V: three nested functions[edit | edit source]

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):

Original (outermost) function
Define as innermost function
as middle function
Express chain rule applicable here
Differentiate f(g)[2]
Differentiate
Differentiate
Substitute into chain rule.

Chain Rule in Physics[edit | edit source]

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 | edit source]

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
  1. Choose coordinate system
  2. Identify variables
  3. Draw picture
  4. Big idea: use chain rule to compute rate of change of distance between two vehicles
  5. Express in terms of and via Pythagorean theorem
  6. Express using chain rule in terms of and
  7. Substitute in
  8. 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  :

Apply derivative operator to entire function
Sum of squares is inside function
Distribute differentiation operator
Apply chain rule to and
Simplify.


Substitute in and simplify

Consequently, the two vehicles are getting closer together at a rate of .

Physics Example II: harmonic oscillator[edit | edit source]

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:

Definition of velocity in one dimension
Substitute
Bring constant outside of derivative
Differentiate outside function (cosine)
Bring negative sign in front
Evaluate remaining derivative
Simplify.

The acceleration is then the second time derivative of position, or simply .

Definition of acceleration in one dimension
Substitute
Bring constant term outside of derivative
Differentiate outside function (sine)
Evaluate remaining derivative
Simplify.

From Newton's second law, , where is the net force and is the object's mass.

Newton's second law
Substitute
Simplify
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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

  1. http://www.math.brown.edu/help/derivtips.html
  2. The derivative of is  ; see Calculus/Derivatives of Exponential and Logarithm Functions.
  3. a b University of British Columbia, UBC Calculus Online Course Notes, Applications of the Chain Rule, http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/chainap.html Accessed 11/15/2010.

External links[edit | edit source]

← Derivatives of Trigonometric Functions Calculus Higher Order Derivatives →
Chain Rule