Step 1: Understand what differentiability means. A function is differentiable at a point if it has a defined slope (derivative) at that point.
Step 2: Identify the function given, which is f(x) = |x|. This function represents the absolute value of x.
Step 3: Determine the point we are interested in, which is x = 0.
Step 4: Calculate the left-hand derivative at x = 0. This means we look at values of x that are slightly less than 0 (like -0.1). The slope is calculated as the limit of (f(0) - f(-0.1)) / (0 - (-0.1)) as we approach 0 from the left.
Step 5: Calculate f(0) = |0| = 0 and f(-0.1) = |-0.1| = 0.1. So, the left-hand derivative is (0 - 0.1) / (0 - (-0.1)) = -0.1 / 0.1 = -1.
Step 6: Now, calculate the right-hand derivative at x = 0. This means we look at values of x that are slightly more than 0 (like 0.1). The slope is calculated as the limit of (f(0.1) - f(0)) / (0.1 - 0) as we approach 0 from the right.
Step 7: Calculate f(0.1) = |0.1| = 0.1. So, the right-hand derivative is (0.1 - 0) / (0.1 - 0) = 0.1 / 0.1 = 1.
Step 8: Compare the left-hand derivative (-1) and the right-hand derivative (1). Since they are not equal, the function is not differentiable at x = 0.
Differentiability – The property of a function to have a derivative at a given point, which requires the left-hand and right-hand derivatives to be equal.
Absolute Value Function – A piecewise function defined as f(x) = x for x ≥ 0 and f(x) = -x for x < 0, which has a corner point at x = 0.
Left and Right Derivatives – The derivatives calculated from the left and right sides of a point, used to determine differentiability.
