1. The problem is to find the derivative of the function $f(x) = 4 - |x|$ at $x=0$. 2. The absolute value function $|x|$ is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ 3. Therefore, $f(x)$ can be written as a piecewise function: $$f(x) = \begin{cases} 4 - x & \text{if } x \geq 0 \\ 4 + x & \text{if } x < 0 \end{cases}$$ 4. To find $f'(0)$, we need to check the left-hand and right-hand derivatives at $x=0$. 5. The right-hand derivative (for $x \to 0^+$) is the derivative of $4 - x$, which is: $$f'_+(0) = -1$$ 6. The left-hand derivative (for $x \to 0^-$) is the derivative of $4 + x$, which is: $$f'_-(0) = 1$$ 7. Since the left-hand and right-hand derivatives at $x=0$ are not equal ($1 \neq -1$), the derivative $f'(0)$ does not exist. 8. In conclusion, the function $f(x) = 4 - |x|$ is not differentiable at $x=0$ because of the sharp corner in the graph. Final answer: $$f'(0) \text{ does not exist}$$