Infinite Derivatives

Situation	Geometric Meaning
f'(x_0) = +\infty	Vertical tangent, curve rises steeply at $x_0$
f'(x_0) = -\infty	Vertical tangent, curve falls steeply at $x_0$
f'_+(x_0) = +\infty, f'_-(x_0) = -\infty	Cusp pointing vertically upward
f'_+(x_0) = -\infty, f'_-(x_0) = +\infty	Cusp pointing vertically downward

Extending the Definition

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So far, saying a function $f$ has a derivative at $x_0$ has meant that f'(x_0) is finite. However, it is useful to broaden the definition of derivatives to encompass scenarios where the limit of the difference quotient ${\displaystyle \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}$ may be infinite. This expansion ensures that the geometric interpretation of the derivative, as the slope of a tangent line, remains applicable even when the tangent line is vertical.

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In cases where the derivative is infinite, we cannot prove $f$ is continuous at $x_0$ from the difference-quotient limit alone. Therefore, we explicitly require continuity at $x_0$ as part of the definition for an infinite derivative to exist. This maintains the relationship between continuity and differentiability while extending the definition of the derivative to include vertical tangent lines through the use of infinite values.

Definition

Suppose that $f$ is continuous at $x_0$. If

$$\frac{f(x_0+\mathrm{\Delta }x)-f(x_0)}{\mathrm{\Delta }x}$$

approaches $+\mathrm{\infty }$ (or $-\mathrm{\infty }$) as $\mathrm{\Delta }x\to 0$, we write f'(x_0) = +\infty (or f'(x_0) = -\infty).

One-Sided Infinite Derivatives and Cusps

We can also define one-sided infinite derivatives. When the one-sided infinite derivatives have opposite signs (as shown in the following figure), there is a unique vertical tangent line. This type of singularity with infinite one-sided derivatives occurs due to the presence of a cusp pointing vertically up or down.

By allowing derivatives to take infinite values, we can account for vertical tangent lines caused by cusps and still retain the geometric interpretation of the derivative as the slope of the tangent line.

Frequently Asked Questions

If f'(x_0) = +\infty, is $f$ differentiable at $x_0$?

It depends on the convention. In the strict definition, differentiability requires the derivative to be a finite number. An infinite derivative is sometimes called an improper derivative. In this extended sense, $f$ has an infinite derivative at $x_0$, but it is not differentiable in the strict sense. The key requirement is that $f$ must be continuous at $x_0$ for an infinite derivative to be defined there.

What is the tangent line when the derivative is infinite?

When f'(x_0) = \pm\infty, the tangent line is a vertical line $x=x_0$. A vertical line has undefined (infinite) slope, which is consistent with the derivative being infinite.

What is a cusp and how does it differ from a corner?

A corner occurs when the left and right one-sided derivatives are both finite but unequal, producing two distinct tangent rays at the point. A cusp occurs when the one-sided derivatives are both infinite with opposite signs, producing a single vertical tangent line. Visually, a cusp looks like a sharp pointed tip, while a corner looks like a bend or kink in the curve.