Differentiability: Everything You Need To Know

• 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:

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

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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.

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