Why does the derivative not exist at a cusp

Consider the function. It is clear that the graph of this function becomes vertical and then virtually doubles back on itself.

Such pattern signals the presence of what is known as a vertical cusp. In general we say that the graph of f x has a vertical cusp at x 0 , f x 0 iff. In both cases, f ' x 0 becomes infinite. A graph may also exhibit a behavior similar to a cusp without having infinite slopes: Example. In calculus, a one-sided limit is either of the two limits of a function f x of a real variable x as x approaches a specified point either from the left or from the right.

Each point in the derivative of a function represents the slope of the function at that point. In the case of a sharp point, the limit from the positive side differs from the limit from the negative side, so there is no limit. The derivative at that point does not exist. A function is not differentiable at a if its graph has a corner or kink at a. Since the function does not approach the same tangent line at the corner from the left- and right-hand sides, the function is not differentiable at that point.

The graph to the right illustrates a corner in a graph. The derivative f' x is the rate of change of the value of function relative to the change of x. All these functions are almost constant around 0, which is the value where their derivatives are 0. So the only possible candidates for the x-coordinate of an extreme point are the critical points and the endpoints.

A cusp, or spinode, is a point where two branches of the curve meet and the tangents of each branch are equal. A tangent of a curve is a line that touches the curve at one point. It has the same slope as the curve at that point. A vertical tangent touches the curve at a point where the gradient slope of the curve is infinite and undefined.

A function is not differentiable where it has a corner, a cusp, a vertical tangent, or at any discontinuity. These are some possibilities we will cover. Examples of corners and cusps. At a cusp, the function is still continuous, and so the limit exists. Here we have a point where our graph peaks out. If we were to take the tangent line at that point we would get a horizontal line.

Recall that the slope of a horizontal line is zero. So for the derivative at that point we plot it at the y-coordinate of the slope of that tangent line. In other words, zero. Notice there's a point to the right where the graph bottoms out. Here, we will also get a slope of zero. So the derivative point will be zero. Now let's look at a point between the peak and bottom.

If we take the tangent line we see it it's decreasing, starting from a higher point and ending at a lower point. This means it's slope is negative. You might notice that at every point between the peak and bottom has a negative slope. It will be the most negative at the point exactly in between the two. As we start from the peak the slope gets more and more negative until it reaches the middle. From there is gets less and less negative until it reaches the the bottom out point where the slope is zero.

On the derivative graph we see:. The more negative the slope of the tangent line, the more negative the y-coordinate of the derivative function.