Differentiability

The derivative of a function exists if the left-hand derivative is equal to the right-hand derivative, just like with limits. There are only a few times that the derivative does not exist, and it is usually narrowed down to a specific point, or set of points.

The derivative fails to exist at a ...

1. Corner, like the one found in y=abs(x). This function is not differentiable at x=0. If you look at the left part of the function (x<0), y="x.">0), the function is described by the equation y= -x. The slope of the tangent line (the derivative) is -1. The two derivatives do not match, so the derivative does not exist at x = 0.

2. Cusp, like the one found in y = x^(2/3). If you look at this graphically you can see that the slope at x = 0 changes very quickly from positive to negative. We will look at the slope of this equation later when we have some more rules for derivatives.

3. Vertical tangent, like the one found in y = cuberoot(x). If you look at the graph of this function you will see that at x = 0, you will have a vertical tangent line. This is where I said you may want to use a small piece of paper to represent the tangent line. When you get around x = 0, your piece of paper will be pointing straight up, indicating a vertical tangent. Again, we will look at calculating this derivative (to the left and right of x =0) later on.

4. Discontinuity. We already studied discontinuities back in chapter 2. As a refresher, they are point (removable), infinite, oscillating, and jump. This means that if we find any discontinuities, our function will also not be differentiable for that particular x-value.

Now I gave you some instances, but it was merely a coincidence that our points where the functions were not differentiable were at x = 0, so don't fall into the trap of automatically thinking that is. Have a great long weekend.