Differentiability: Everything You Need To Know

Differentiability is a fundamental concept in calculus and has significant implications in various fields, such as physics, engineering, and economics. Understanding where a function is differentiable, and how its derivatives behave, is crucial in the study of rates of change and the behavior of functions. The examples provided demonstrate different scenarios of differentiability and non-differentiability, illustrating the practical application of this concept in mathematical analysis.

• Differentiability and Smoothness: A function is differentiable at a point if it is “smooth” (without sharp corners or cusps) and continuous at that point. The smoothness implies that the function’s graph can be approximated by a tangent line at every point where the function is differentiable.
• Differentiability and Continuity: While differentiability implies continuity, the converse isn’t always true. A function can be continuous but not differentiable at certain points. A typical example is the absolute value function \( f(x) = |x| \), which is continuous everywhere but not differentiable at \( x = 0 \) due to a sharp corner.

Examples:

1. Quadratic Function:

\( f(x) = x^2 \Rightarrow f'(x) = 2x \)

Since \( f'(x) \) is defined and continuous for all \( x \), \( f(x) \) is differentiable everywhere.

2. Absolute Value Function (Continuous but not Differentiable at a Point):

\( f(x) = |x| \)

\( f'(x) \) does not exist at \( x = 0 \) because of the sharp corner in its graph.

\( f(x) \) is continuous everywhere but not differentiable at \( x = 0 \).

3. Trigonometric Function:

\( f(x) = \sin x \Rightarrow f'(x) = \cos x \)

Since \( f'(x) \) is continuous and defined for all \( x \), \( f(x) \) is differentiable everywhere.

4. Piecewise Function (Continuity Without Differentiability):

\( f(x) = \begin{cases} x^2 & \text{if } x \leq 0 \ \sqrt{x} & \text{if } x > 0 \end{cases} \)

The derivative changes abruptly at \( x = 0 \). Although \( f(x) \) is continuous at \( x = 0 \), it is not differentiable there due to the abrupt change in the rate of change.

Another Example:

\( f(x) = x^3 \text{ and } g(x) = x^{1/3}. \)

\( f(x) = x^3, \ f'(x) = 3x^2 \) \\

\( \text{Since } f'(x) \text{ is continuous and defined for all } x, \ f(x) \text{ is differentiable everywhere.} \)

\( g(x) = x^{1/3}, \ g'(x) = \frac{1}{3}x^{-2/3} \) \\

\( \text{Since } g'(x) \text{ is not defined at } x = 0, \ g(x) \text{ is not differentiable at } x = 0. \)

\( g(x) = x^{2/3}, \ g'(x) = \frac{2}{3}x^{-1/3} \) \\

\( \text{Since } g'(x) \text{ is not defined at } x = 0, \ g(x) \text{ is not differentiable at } x = 0. \)

If a function is not differentiable at a certain point, it means you cannot calculate its derivative at that point. The function lacks a defined rate of change or slope at that specific location on its graph.