Understanding Why Derivatives Fail at Vertical Tangents
Derivative essentially means slope, which is a fundamental concept in calculus. However, when a curve has a vertical tangent, the slope becomes undefined, leading to the failure of the derivative at that point.
The Concept of Slope and Derivative
As you know, a derivative is the slope of a curve at a given point. The slope of a vertical line is undefined because the angle between a vertical line and the x-axis is 90 degrees. Since tan(90°) is undefined, the derivative at a vertical tangent is also undefined.
Derivative and Vertical Tangents
The angle between the tangent and the x-axis at a point on a curve is called the angle of inclination. For a vertical tangent, this angle is 90 degrees. The derivative at such a point, which is the tangent of this angle, is undefined. Hence, the derivative fails to exist at a vertical tangent.
Handling Infinity in Derivatives
It is important to note that the definition and handling of infinity in derivatives depend on the approach. For example, when computing the derivative of the cube root of x at x 0, using the definition of the derivative, one gets infinity. Most calculus textbooks classify this as an undefined derivative. However, in many applications, infinity is meaningful—it indicates an infinitely steep slope, i.e., a vertical tangent.
When teaching calculus, particularly for students in fields of application, it is often useful to extend the notion of the derivative to include positive and negative infinity. This allows for a more comprehensive understanding of the behavior of functions with vertical tangents.
Limits and Derivative Existence
A limit does not exist if it does not converge to a finite value. In the context of derivatives, if the limit used to compute the derivative does not yield a finite value, we say the derivative does not exist. For example, at a vertical tangent, the derivative does not take a finite value and hence is said to be undefined or the function is not differentiable at that point.
Vertical Tangents and Functions
A vertical tangent is a special case where the curve appears to be tangent to a vertical line. This is different from the concept of a tangent in geometry. In geometry, a tangent is a line that touches a curve at a single point without crossing it. In calculus, a function with a vertical tangent does not exist in the conventional sense because the y-value is not uniquely determined by the x-value at that point.
For instance, consider the function $f(x) sqrt[3]{x}$ at x 0 or the function $f(x) sqrt[2]{|x|}$. At x 0, these functions have vertical tangents. However, since the derivative is not defined due to the limit not converging to a finite value, these functions are not differentiable at x 0.
In general, for a function to have a derivative at a point, it must satisfy certain conditions. If the limits of the left-sided and right-sided differential quotients at a point are not equal, the function is not differentiable at that point. Similarly, if the function is not defined on one side of the point, it cannot be differentiable at that point either.
Conclusion
Understanding why derivatives fail at vertical tangents is crucial in calculus. These tangents indicate a significant change in the behavior of the function and highlight the importance of considering the limits and definitions of derivatives carefully. By expanding our understanding of these concepts, we can better analyze and work with complex functions.