Differentiability

Because the derivative is a limit, it may not exist. It exists if and only if \[\lim\limits_{h\to0^-}\dfrac{f(x+h)-f(x)}{h}=\lim\limits_{h\to0^+}\dfrac{f(x+h)-f(x)}{h}\]

Furthermore, the derivative is undefined if the limit exists but is infinite, which means the tangent to the graph is vertical. Since the slope of the tangent is equivalent to the derivative, it is undefined in this case.

The derivative of a function is undefined in the following cases.

  • At points of discontinuity, whether the function is defined at those points or not. If the function is undefined, it is easy to see why, because \(f(x)\) is part of the definition of the derivative. In the graphs below, drag the slider to change \(h\) and notice how the secants change from the left and the right.
    In point discontinuity, the secants from both sides approach a vertical line. This function is indifferentiable at \(x=0\).
    In jump discontinuity, the secant from at least one side approaches a vertical line. This function is indifferentiable at \(x=0\).
  • At a corner, where the tangents from the left and the right have different slopes.
    At a corner, the slopes of the tangents from the left and the right are different. This function is indifferentiable at \(x=1\).
  • At a cusp, where the tangents from the left and the right have slopes with the same absolute value but different signs.
    At a cusp, the slopes of the tangents from the left and the right have the same absolute value but different signs. This function is indifferentiable at \(x=1\).
  • Wherever the graph of a function has a vertical tangent.
    In this continuous function, the secant approaches a vertical line as \(x\to 0\). The secants from the left and the right have the same slope, but this function is indifferentiable at \(x=0\).
  • At the boundaries of the domain if the domain is a closed interval. This is because we can approach the boundary point from one side, but not from the other.

The domain of the derivative of a function is never less restricted than that of the original function. It may only have more restrictions.