Here's a fun (but untrue) fact.
You know from calculus that the derivative of x^{2}
is 2x. But what's wrong with the following calculation?
x^{2} = x + x + ... + x (repeated x times)
so by taking the derivative of both sides we get
(x^{2})' = 1 + 1 + ... + 1 = x.
Hmmn...
Presentation Suggestions:
This is a great Fun Fact to use to point out to students
who already think they know calculus that there may
still be a gap in understanding!
The Math Behind the Fact:
Fallacious arguments such as this one help to elicit
understanding. The argument above breaks down because
we took the derivative of x different x's.
So each term depends on x and we accounted for this
when we took the derivative, but also
the number of terms (which could be
fractional) also depends on x, and this
was not accounted for. Put another way, the derivative
measures the rate of change of (x^{2}) as
x changes, but as x changes, the number of terms on the
right as well as the terms themselves increase. So
for positive x, the "right" answer should be larger than x,
and it is, indeed, 2x.
How to Cite this Page:
Su, Francis E., et al. "Derivative Paradox."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
