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