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

Problem

d(x−2)/d(x)

Solution

1. Identify the function as a composition involving the absolute value function, ƒ(x)=|u| where u=x−2

2. Recall the derivative rule for the absolute value function, which is d(u)/d(u)=u/|u| for u≠0

3. Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function u=x−2

4. Calculate the derivative of the inner function, which is d(x−2)/d(x)=1

5. Combine the results to find the derivative of the entire expression.

6. Note that the derivative is undefined at x=2 because the function has a sharp corner (cusp) at that point.

Final Answer

d(x−2)/d(x)=(x−2)/|x−2|

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