similar to |x| wouldn't x^2 not be differenciable?

The function \( |x| \) is not differentiable at \( x = 0 \) because it has a sharp corner at that point, where the left-hand derivative and right-hand derivative are different. However, this is not the case for \( x^2 \).

Here's a breakdown of why \( x^2 \) is differentiable:

- **For \( |x| \):**
  - \( |x| \) is defined as:
    \[
    |x| = \begin{cases} 
    x, & \text{if } x \geq 0 \\
    -x, & \text{if } x < 0
    \end{cases}
    \]
  - At \( x = 0 \), the left-hand derivative is \( -1 \), and the right-hand derivative is \( 1 \), so the derivative does not exist at \( x = 0 \).

- **For \( x^2 \):**
  - The function \( x^2 \) is smooth and differentiable everywhere.
  - The derivative of \( x^2 \) is \( 2x \), and at \( x = 0 \), this derivative is \( 2(0) = 0 \).
  - Since \( x^2 \) is a smooth curve (a parabola), there are no sharp corners or discontinuities, meaning it's differentiable at all points, including \( x = 0 \).

In conclusion, \( |x| \) is not differentiable at \( x = 0 \) because of the corner, but \( x^2 \) is differentiable everywhere, including at \( x = 0 \), because it has a smooth slope.

Would you like more details on differentiability or any related topic?

Here are 5 related questions:
1. What is the difference between continuous and differentiable functions?
2. How do you determine if a piecewise function is differentiable at a point?
3. Can a function be continuous but not differentiable?
4. What are other examples of functions that are not differentiable at certain points?
5. What is the geometric interpretation of differentiability?

**Tip:** A function is differentiable at a point if it has a unique, well-defined tangent (no sharp turns or cusps) at that point.

similar to |x| wouldn't x^2 not be differentiable?

Explore the differentiability of |x| and x^2, and why |x| has a sharp corner at x = 0 while x^2 is smooth and differentiable everywhere. This guide covers key concepts such as absolute value functions, quadratic functions, and their derivatives. Ideal for students in Grades 10-12.

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