Question

If \( f(x) = x|x| \), then \( f'(-1) + f'(1) \) is equal to:

• A. 2
• B. -2
• C. 0
• D. -4
• E. 4

Hint:

For piecewise functions, differentiate each piece separately based on the domain of the function.

Answer 1

Answer: (E)

Step 1: The function \( f(x) = x|x| \) can be written as: \[ f(x) = \left\{ \begin{array}{ll} x^2, & x \geq 0 \\ -x^2, & x < 0 \end{array} \right. \] 
Step 2: Differentiating \( f(x) \) piecewise: For \( x > 0 \), \( f'(x) = 2x \). For \( x < 0 \), \( f'(x) = -2x \). 
Step 3: Substitute \( x = -1 \) and \( x = 1 \): \[ f'(-1) = -2(-1) = 2, \quad f'(1) = 2(1) = 2. \] 
Step 4: Therefore: \[ f'(-1) + f'(1) = 2 + 2 = 4. \]