related rates

The notion of a derivativeMathworldPlanetmath has numerous interpretationsMathworldPlanetmathPlanetmath and applications. A well-known geometric interpretation is that of a slope, or more generally that of a linear approximation to a mapping between linear spacesPlanetmathPlanetmath (see Another useful interpretation comes from physics and is based on the idea of related rates. This second point of view is quite general, and sheds light on the definition of the derivative of a manifold mapping (the latter is described in the pushforward entry).

Consider two physical quantities x and y that are somehow coupled. For example:

  • the quantities x and y could be the coordinatesMathworldPlanetmathPlanetmath of a point as it moves along the unit circle;

  • the quantity x could be the radius of a sphere and y the sphere’s surface areaMathworldPlanetmath;

  • the quantity x could be the horizontal position of a point on a given curve and y the distanceMathworldPlanetmath traversed by that point as it moves from some fixed starting position;

  • the quantity x could be depth of water in a conical tank and y the rate at which the water flows out the bottom.

Regardless of the application, the situation is such that a change in the value of one quantity is accompanied by a change in the value of the other quantity. So let’s imagine that we take control of one of the quantities, say x, and change it in any way we like. As we do so, quantity y follows suit and changes along with x. Now the analytical relationMathworldPlanetmath between the values of x and y could be quite complicated and non-linear, but the relation between the instantaneous rates of change of x and y is linear.

It does not matter how we vary the two quantities, the ratio of the rates of change depends only on the values of x and y. This ratio is, of course, the derivative of the function that maps the values of x to the values of y. Letting x˙,y˙ denote the rates of change of the two quantities, we describe this conception of the derivative as


or equivalently as

y˙=dydxx˙. (1)

Next, let us generalize the discussion and suppose that the two quantities 𝐱 and 𝐲 represent physical states with multiple degrees of freedom. For example, 𝐱 could be a point on the earth’s surface, and 𝐲 the position of a point 1 kilometer to the north of 𝐱. Again, the dependence of 𝐲 and 𝐱 is, in general, non-linear, but the rate of change of 𝐲 does have a linear dependence on the rate of change of 𝐱. We would like to say that the derivative is precisely this linear relation, but we must first contend with the following complication. The rates of change are no longer scalars, but rather velocity vectors, and therefore the derivative must be regarded as a linear transformation that changes one vector into another.

In order to formalize this generalized notion of the derivative we must consider 𝐱 and 𝐲 to be points on manifolds X and Y, and the relation between them a manifold mapping ϕ:XY. A varying 𝐱 is formally described by a parameterized curve


The corresponding velocities take their value in the tangent spaces of X:


The “coupling” of the two quantities is described by the compositionMathworldPlanetmath


The derivative of ϕ at any given 𝐱X is a linear mapping


called the pushforward of ϕ at 𝐱, with the property that for every trajectory γ passing through 𝐱 at time t, we have


The above is the multi-dimensional and coordinate-free generalizationPlanetmathPlanetmath of the related rates relation (1).

All of the above has a perfectly rigorous presentation in terms of manifold theory. The approach of the present entry is more informal; our ambition was merely to motivate the notion of a derivative by describing it as a linear transformation between velocity vectors.

Title related rates
Canonical name RelatedRates
Date of creation 2013-03-22 12:44:59
Last modified on 2013-03-22 12:44:59
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 6
Author rmilson (146)
Entry type Definition
Classification msc 53A45
Classification msc 53A17
Classification msc 26A24
Related topic Derivative2