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19 September 2003

The Sum of Derivatives

by George E. Hrabovsky, President, MAST

Where We Have Been

Last week I made a mistake in the Math Challenge, I asked you to prove the theorems when there weren't any.  I also said that I gave the formal definition of the limit, when I actually gave the formal definition of the derivative.  My mistakes! Sorry! There is no excuse for such sloppiness.

Another Problem in the Lab

From last time, let's say that you have analyzed some data and determined that it forms a pattern based on the variable time, t,

FormBox[RowBox[{x, , =, , RowBox[{t^3, , +, , 34 t^2, , -, , 127 t, , -, , 56.}]}], TraditionalForm]

A colleague asks, "What is the time derivative of this expression?"

From last time we know that the result is

FormBox[RowBox[{x/t, , =, , RowBox[{Underscript[lim, Δ t0] ( ... t), =, , RowBox[{3 t^2,   , +, 68 t,   , -, , 127.}]}]}]}], TraditionalForm]

This was arrived at after a good deal of algebra.  Is there a simpler way to arrive at this result?  If we look at the algebra, we have

change

          = ((t + Δ t)^3 + 34 (t + Δ t)^2 - 127 (t + Δ t) - 56 - t^3 - 34 t^2 + 127 t + 56)/(Δ t) .

If we stare at this long enough we realize that we have

change = ((t + Δ t)^3 - t^3)/(Δ t) + (34 (t + Δ t)^2 - 34 t^2)/(Δ t) - (127 (t + Δ t) - 127 t)/(Δ t) - (56 - 56)/(Δ t) .

This looks familiar.  By theorem 1 of "Putting Limits Together," we have the limit of a sum or difference being equal to the sum or difference of the limits.  The same must be true of derivatives, since the derivative is a limit.  So we conjecture then that the derivative of a sum is the sum of the derivatives. We write,

/t [f(t) g(t)] = /t f(t) /tg(t) .

A shorter way of denoting a derivative is f ' (t),

/t [f(t) g(t)] =   f ' (t) g ' (t) .

The Sum Rule

What we have just conjectured is one of the most fundamental rules of differentiation, it is called the sum rule.

/t [f(t) g(t) ...] =   f ' (t) g ' (t) ... .

The Math Challenge from Last Time

There was no real math challenge last time.

The Math Challenge

Can you prove the sum rule?  Can you define a derivative in terms of ϵ - δ notation?

Math Resources for Derivatives

Online:

http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/

This is a neat site.  For a more challenging page look at this one:

http://www.mathpages.com/home/icalculu.htm

Both sites are very cool.

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a rigorous, but readable chapter on derivatives.

Created by Mathematica  (September 18, 2003)