Question:
Let \( f(x)=|x-2| \) and \( g(x)=f(f(x)) \). Then derivative of \( g \) at the point \( x=5 \) is
Let \( f(x)=|x-2| \) and \( g(x)=f(f(x)) \). Then derivative of \( g \) at the point \( x=5 \) is
Show Hint
Always evaluate modulus cases before differentiating.
Updated On: Apr 30, 2026
- \(1\)
- \(2\)
- \(4\)
- \(5\)
- \(0\)
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The Correct Option is A
Solution and Explanation
Concept:
For composite functions involving modulus, evaluate regions carefully and then apply chain rule.
Step 1: Compute inner function. \[ f(5)=|5-2|=3 \]
Step 2: Compute outer function. \[ g(5)=f(3)=|3-2|=1 \]
Step 3: Differentiate using chain rule. \[ g'(x)=f'(f(x)) \cdot f'(x) \]
Step 4: Determine derivatives. For \(x>2\): \[ f'(x)=1 \] Since \(f(5)=3>2\): \[ f'(3)=1 \]
Step 5: Compute derivative. \[ g'(5)=1 \cdot 1 = 1 \]
Step 1: Compute inner function. \[ f(5)=|5-2|=3 \]
Step 2: Compute outer function. \[ g(5)=f(3)=|3-2|=1 \]
Step 3: Differentiate using chain rule. \[ g'(x)=f'(f(x)) \cdot f'(x) \]
Step 4: Determine derivatives. For \(x>2\): \[ f'(x)=1 \] Since \(f(5)=3>2\): \[ f'(3)=1 \]
Step 5: Compute derivative. \[ g'(5)=1 \cdot 1 = 1 \]
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