Derivative, slope or, the tangent of a graph with a shape which has corner like tips as in the letter V

Question

Consider the graph of a function in the shape of the letter ‘V’, how would we be finding the derivative, slope or, the tangent of the function at the value of the function that corresponds to the tip of ‘V’?

Answer 1

This kind of "tip" is called a cusp, and to see why the derivative is not defined at a cusp, look at the instantaneous slopes on either side of the cusp.

As you approach from either side, the slope will approach different values, and because the derivative is defined as a limit and limits only exist when they approach the same value from both sides, the derivative doesn't exist at the cusp.

Answer 2

The derivative isn’t defined at the tip (or, in other words, the function is not differentiate at that point).

There are concepts like semi-differentiable, left derivative, and right derivative, which can be applied to functions whose graphs have sharp corners.

Answer 3

We can use sub-gradients as shown in the following figure, taken from here. This method particularly useful for backpropagation of neural nets in machine learning (e.g., RELU activation function, which is not differentiable everywhere).