<\/span><\/h2>\nNo, the absolute value function is not differentiable at points where its derivative is undefined.<\/strong> The absolute value function has a kink at x=0, which is the point where it is not differentiable. At this point, the derivative does not exist because the slope of the function abruptly changes.<\/p>\nIn order to understand why the absolute value function is not differentiable at x=0, let’s take a closer look at its graph. The graph of |x| consists of two linear segments – one with a positive slope on the right side of the y-axis, and another with a negative slope on the left side of the y-axis. They intersect at the origin, creating a sharp point or kink.<\/p>\n
At any point to the left of zero, the slope of the graph becomes negative, whereas the slope is positive to the right of zero. However, at x=0, the slope cannot be precisely defined as it undergoes an abrupt change. This discontinuity at x=0 results in the absolute value function being non-differentiable at this point.<\/p>\n