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20 February 2004

The Chain Rule

by George Hrabovsky

by George E. Hrabovsky, President, MAST

What We Did Last Time

We explored the general first-order linear homogeneous differential equations.

Where We Will Go This Time

I will present a useful technique for differentiation.

Differentiation of Composite Functions

We have been talking about differential equations. Before we go much further, we will need to develop some special tools.  For example, how do we take the derivative of the function of a function?  Let us say that we have

f(x) = x^2

and

g(f(x)) = sin f(x) .

To proceed we use what is called the chain rule,

?g(x)/?x = ?g(x)/?f(x) ?f(x)/?x .

In our case, we have,

?g(x)/?x = ?/?f(x)[sin f(x)] ?/?x (x^2)

                = cos f(x) 2 x

                = 2 x cos x^2 .

Here is another example,

a(t) = ?^t

and

b(a(t)) = a(t)^3/(tan a(t)) .

So,

?b(t)/?t = ?/?a(t)[a(t)^3/(tan a(t))] ?/?t (?^t)

               =[3 a(t)^2 cot a(t) - a(t)^3 csc^2 a(t)] ?^t

               = 3 ?^(3 t) cot ?^t - ?^(4 t) csc^2 ?^t .

This is a powerful tool.

Mathematics Challenge from Last Time

The problem was how to take the derivative of a function of more than one variable.  You can take the derivative of each variable separately; this is called a partial derivative . For example, if we have a function of two variables,

f(x, y) = x^2 + 2 y^2,

we could take two partial derivatives (note the special symbol for partial derivatives)

?f/?x = ?/?x (x^2 + 2 y^2)

          = 2x .

and

?f/?y = ?/?y (x^2 + 2 y^2)

          = 4y .

Mathematics Challenge

Speculate on how the chain rule relates to partial derivatives.

Sources About Differential Equations

On-Line:

The ultimate resource for those interested in research related to differential equations is the Electronic Journal of Differential Equations, which is free:

http://ejde.math.swt.edu/

Another nice site that covers lots of topics, though with only a few examples, is the SOS Mathematics page for differential equations:

http://www.sosmath.com/diffeq/diffeq.html

Another free online journal is the Electronic Journal of Qualitative Theory of Differential Equations:

http://www.math.u-szeged.hu/ejqtde/

An online course in differential equations is given at this site:

http://math.stcc.edu/DiffEq/DiffEq.html

Books:

Martin Braun (1993), Differential Equations and Their Applications , Springer-Verlag.  This is a nice book crammed full of applications.

George F. Simmons (1991), Differential Equations with Applications and Historical Notes, McGraw-Hill.  This is one of my favorite differential equations books, it starts out at an elementary level and includes nonlinear equations and partial differential equations.  My columns will loosely follow this book up to a point and will then take a more geometrical viewpoint.

John W. Dettman (1974), Introduction to Linear Algebra and Differential Equations, McGraw-Hill (reprinted in 1986 by Dover Publications). This is a wonderful book that is also inexpensive; it begins with a discussion of complex numbers, then linear algebra, and then differential equations.

Morris Tenenbaum and Harry Pollard (1963), Ordinary Differential Equations, Harper and Row (reprinted in 1985 by Dover Publications). This is an inexpensive and encyclopedic treatment of the subject.

Vladimir I. Arnold (1973), Ordinary Differential Equations, MIT Press (as of 1991 in its 8th printing, translated from the Russian by Richard A. Silverman).  This is my favorite book of differential equations at a fairly elementary level.  What makes this book so remarkable is the use of geometric methods.  My columns will follow this book after a firm foundation has been laid with traditional methods.

Created by Mathematica  (February 19, 2004)