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26 September 2003

Three More Rules

by George E. Hrabovsky, President, MAST

Where We Have Been

We have been exploring derivatives. Last time we developed the sum rule.

The Same Problem in the Lab

From last time we developed the expression

x/t = ((t + Δ t)^3 - t^3)/(Δ t) + (34 (t + Δ t)^2 - 34 t^2)/(&# ... nbsp; ]}]}], (1)}]}], TraditionalForm]

Also from last time, we derived the rule,

So, we can rewrite (1)

$FormBox[RowBox[{x/t, , =, RowBox[{/t t^3, +, / ... /t 127 t, , -, RowBox[{/t, 56.}]}]}], TraditionalForm]$

Now we just do the differentiation for each term. Let's take the simplest term first, the last term in the expression is the derivative of a constant. A constant, by definition, never changes, so the derivative is always 0. We can write

So, we can rewrite (1)

The next to the last term is a product. If we look generally at the product of two functions of in a table of derivatives we find

In our case the second term is

By the constant rule the last term must be 0,

So, we can rewrite (1),

$FormBox[RowBox[{x/t, , =, RowBox[{/t t^3, +, /t34 t^2, , -, , 127.}]}], TraditionalForm]$

The next term is a product involving a power. If we look in a table of derivatives, we see

In our case we have,

So, we can rewrite (1)

$FormBox[RowBox[{x/t, , =, RowBox[{/t t^3, +, 68 t, , -, , 127.}]}], TraditionalForm]$

We can also apply the power rule to the very first term,

Thus, we can rewrite (1)

$FormBox[RowBox[{x/t, , =, RowBox[{3 t^2, +, 68 t, , -, , 127.}]}], TraditionalForm]$

This is the time derivative of the function,

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

The Constant Rule

We have just conjectured three more rules that are fundamental to differentiation. The first is called the constant rule.

The Product Rule

The second rule we have conjectured is called the product rule.

The Power Rule

The third rule we have conjectured is called the power rule.

The Math Challenge from Last Time

I posed two challenges last time. The first was to prove the sum rule. We have, for functions of ,

We can then use mathematical induction to show that this works for any number of sums we care to add.

The second task was to write the definition for the derivative in notation. We have,

For any there exists such that | (f(t + Δ t) - f(t))/(Δ t) - /tf(t) | <ϵ whenever .

The Math Challenge

Can you prove the three rules presented today?

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 25, 2003)