Derivative of y = |x| Graph the derivative of f(x) = |x|. Then graph y = (|x| − 0)/(x − 0) = |x|/x. What can you conclude?

Verified step by step guidance

1. To find the derivative of y = |x|, consider the piecewise definition of the absolute value function: y = x for x >= 0 and y = -x for x < 0.
2. Differentiate each piece separately: For x > 0, the derivative of y = x is 1. For x < 0, the derivative of y = -x is -1. At x = 0, the derivative is undefined because the function has a sharp corner.
3. Graph the derivative: For x > 0, plot a horizontal line at y = 1. For x < 0, plot a horizontal line at y = -1. At x = 0, indicate that the derivative is undefined, often shown as an open circle or a gap.
4. Now, consider the function y = |x|/x. This is also a piecewise function: y = 1 for x > 0 and y = -1 for x < 0. At x = 0, the function is undefined.
5. Graph y = |x|/x: For x > 0, plot a horizontal line at y = 1. For x < 0, plot a horizontal line at y = -1. At x = 0, indicate that the function is undefined. Notice that the graph of the derivative of |x| matches the graph of |x|/x, except at x = 0 where both are undefined.

Key Concepts

Derivative of Absolute Value Function

The derivative of the absolute value function, y = |x|, is not defined at x = 0 because the function has a sharp corner at this point. For x > 0, the derivative is 1, and for x < 0, the derivative is -1. This discontinuity at x = 0 is crucial for understanding the behavior of the derivative graph.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. For f(x) = |x|, it can be expressed as f(x) = x for x ≥ 0 and f(x) = -x for x < 0. Understanding this helps in graphing and analyzing the derivative, as it shows how the function behaves differently on either side of x = 0.

Graphing Rational Functions

Graphing the function y = |x|/x involves understanding how the numerator and denominator affect the graph. This function is defined for all x ≠ 0 and equals 1 for x > 0 and -1 for x < 0, creating a step function. Recognizing this helps in visualizing the discontinuity and the behavior of the function around x = 0.