Find the Derivative - d/dx f(x)=|x-2|

Problem

d(x−2)/d(x)

Solution

1. Identify the function as a composition of the absolute value function and a linear function. The derivative of |u| is u/|u| or |u|/u for u≠0

2. Apply the chain rule to the expression |x−2| Let u=x−2

3. Differentiate the outer function |u| with respect to u which gives u/|u|

4. Differentiate the inner function u=x−2 with respect to x which gives 1

5. Combine the results using the chain rule formula d(ƒ)/d(x)=d(ƒ)/d(u)⋅d(u)/d(x)

6. Substitute x−2 back in for u

7. Note the domain of the derivative. The derivative does not exist at x=2 because the function has a sharp corner (cusp) there.

Final Answer

d(x−2)/d(x)=(x−2)/|x−2|

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