#3 What is the derivative of f(x) = |x^2 - x| at x = 2.

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justaguide eNotes educator| Certified Educator

The derivative of f(x) = |x^2 - x| at x = 2 has to be determined.

For values of x greater than or equal to 0. f(x) = |x^2 - x| = x^2 - x.

As 2 is greater than 0, f(x) = |x^2 - x| at x = 2 is equivalent to f(x) = x^2 - x

f'(x) = 2x - 1

Substituting x = 2, 2*2 - 1 = 3

The derivative of f(x) = |x^2 - x| at x = 2 is 3.

PS. You seem to have a problem with determining the derivative of functions related to absolute value of the variable. The function f(x) = |x| is continuous for all values of x. It is not differentiable at x = 0. Though this is not the case for powers of the absolute value function like in #1. Please have a look at the following link.

embizze eNotes educator| Certified Educator

`x^2-x>=0 ` for `x<=0,x>=1 ` so `|x^2-x|=x^2-x ` on those intervals.

This does not change the answer, but `|x^2-x| != x^2-x ` for all x>0; e.g. try x=1/2.

When doing problems with absolute value, return to the definition where |x|=-x for x<0, |0|=0, and |x|=x for x>0 where x can be any expression.

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