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Using the Definition to Compute the Derivative - Exercise 2

**Exercise 2.** Discuss the differentiability of

**Answer.** May be the scariest thing about this function is
the absolute value. So the best thing to do is to look for ways
to remove it. Therefore we are led to find out when *x*^{2} - *x* is
positive or negative. We get

Clearly the derivative exists at every point, except maybe at 0
and 1. Let us discuss these two points. Let us start with 0. We
have

Since the function is defined differently from the left and the
right of 0, then we have to consider the limits to the left and
to the right at 0. We have

and

This implies that *f*'(0) does not exist. Similar computations
will also give

and

which implies that *f*'(1) does not exist.

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