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From the Fun Fact files, here is a Fun Fact at the Easy level:

Derivative Paradox

Here's a fun (but untrue) fact.

You know from calculus that the derivative of x2 is 2x. But what's wrong with the following calculation?

x2 = x + x + ... + x (repeated x times)

so by taking the derivative of both sides we get

(x2)' = 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 (x2) 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>.

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Keywords:    paradox, calculus false proof
Subjects:    calculus, analysis
Level:    Easy
Fun Fact suggested by:   Arthur Benjamin
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