Final answer:
To compute the right-hand and left-hand derivatives as limits and check differentiability at a point, we calculate the limits of the slopes of secant lines approaching the point from the right and left. If these limits exist and are equal, the function is differentiable at that point.
Step-by-step explanation:
To compute the right-hand and left-hand derivatives, we first need to determine the limit as we approach the point p from the right and the limit as we approach the point p from the left. Let's denote the function as f(x).
The right-hand derivative, denoted as f'(p+), is calculated by taking the limit as x approaches p from the right of the slope of the secant line connecting the points (p, f(p)) and (p+h, f(p+h)). In the limit, h approaches 0.
The left-hand derivative, denoted as f'(p-), is calculated by taking the limit as x approaches p from the left of the slope of the secant line connecting the points (p-h, f(p-h)) and (p, f(p)). In the limit, h approaches 0.
If both the right-hand and left-hand derivatives exist and are equal, i.e., f'(p+) = f'(p-), then the function is differentiable at the point p.