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.

Contents

ExamplesEdit

Example IEdit

Suppose that a mountain climber ascends at a rate of 0.5 kilometer per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6°C per kilometer. To calculate the decrease in air temperature per hour that the climber experiences, one multiplies 6°C per kilometer by 0.5 kilometer per hour, to obtain 3°C per hour. This calculation is a typical chain rule application.

Example IIEdit

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 IIIEdit

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 valueEdit

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 functionsEdit

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 PhysicsEdit

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 vehiclesEdit

 
One vehicle is headed north and currently located at (0,3); the other vehicle is headed west and currently located at (4,0). 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 oscillatorEdit

 
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 ChemistryEdit

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 LawEdit

 
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,   ,   , and   , 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 GasesEdit

 
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 ruleEdit

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

 

ExercisesEdit

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

ReferencesEdit

  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 linksEdit