Learn how to find the derivative of abs(x). The derivative is d/dx[abs(x)].
Below is the graph of f(x) = abs(x) and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
To differentiate abs(x), we need to consider its piecewise structure. The function is defined as x when x ≥ 0, and -x when x < 0. This requires treating each part separately for differentiation.
💡 Why this works: The Power Rule applies to both pieces of the function separately. For x ≥ 0, abs(x) is x, and for x < 0, abs(x) is -x. Differentiating both gives us the final form for the derivative.
Apply the Power Rule to both parts of abs(x). For x ≥ 0, the derivative of x is 1. For x < 0, the derivative of -x is -1.
💡 Why this works: Therefore, the derivative d/dx[abs(x)] is 1 for x > 0, -1 for x < 0, and undefined at x = 0, where the function has a cusp.
The derivative d/dx[abs(x)] is simplified to a piecewise function: 1 for x > 0, -1 for x < 0, and undefined at x = 0. This reflects the nature of abs(x) changing direction at x = 0.
💡 Why this works: The final form is consistent with the fact that abs(x) has a sharp corner at x = 0, where the derivative does not exist.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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