Learn how to find the derivative of |x|. The derivative is d/dx[|x|].
Below is the graph of f(x) = |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.
Recognize that |x| is a piecewise function, defined differently for positive and negative values of x. For x > 0, |x| = x, and for x < 0, |x| = -x. The Power Rule applies to these individual pieces as they are simple power functions of x.
💡 Why this works: Since the Power Rule applies to functions of the form x^n, where n is a constant, we treat the positive and negative portions of |x| separately. For x > 0, the Power Rule gives us d/dx[x] = 1, and for x < 0, the derivative of -x is d/dx[-x] = -1.
The derivative of |x| involves applying the Power Rule to each piece of the function. For x > 0, d/dx[|x|] = d/dx[x] = 1. For x < 0, d/dx[|x|] = d/dx[-x] = -1.
💡 Why this works: By using the Power Rule, we differentiate the two parts of the piecewise function, giving us a derivative of 1 for x > 0 and -1 for x < 0. The function is not differentiable at x = 0 because there is a discontinuity in the slope at this point.
After applying the rules, we conclude that the derivative of |x| is 1 for x > 0 and -1 for x < 0. At x = 0, the derivative does not exist.
💡 Why this works: The result reflects the nature of the absolute value function, which has a sharp corner at x = 0. The non-differentiability at this point can be verified by observing the sudden change in slope as you approach 0 from either side.
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❌ 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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