Find the Derivative - d/dx f(x)=|x|

Problem

d(x)/d(x)

Solution

1. Identify the definition of the absolute value function ƒ(x)=|x| which is defined piecewise as x for x>0 and −x for x<0

2. Differentiate the function for the case where x>0 which yields d(x)/d(x)=1

3. Differentiate the function for the case where x<0 which yields (d(−)*x)/d(x)=−1

4. Note that the derivative does not exist at x=0 because the left-hand and right-hand limits of the difference quotient are not equal.

5. Express the derivative using the signum function sgn(x) or the equivalent algebraic expression x/|x|

Final Answer

d(x)/d(x)=x/|x|

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