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 .
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. 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.
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.
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):
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
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.
Choose coordinate system
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
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
Substitute in and simplify
Consequently, the two vehicles are getting closer together at a rate of .
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.
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. 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 Gases
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. 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