Learn how to find the derivative of sign(x). The derivative is d/dx[sign(x)].
Below is the graph of f(x) = sign(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.
Sign(x) is a piecewise function that outputs +1, 0, or -1 depending on the value of x. The Power Rule is applied because sign(x) is defined using simple exponentiation for positive and negative inputs.
💡 Why this works: We apply the Power Rule because sign(x) behaves similarly to a function raised to an exponent for positive and negative values. Since sign(x) is constant for both x > 0 and x < 0, the Power Rule tells us that the derivative is zero for these intervals.
Using the Power Rule, the derivative of the constant +1 (for x > 0) and -1 (for x < 0) results in zero in these regions. At x = 0, the derivative is undefined.
💡 Why this works: For x > 0 and x < 0, the function is constant, meaning the derivative is zero. At x = 0, sign(x) has a jump discontinuity, causing the derivative to be undefined at that point.
We arrive at the conclusion that the derivative of sign(x) is zero for x > 0 and x < 0, with the derivative being undefined at x = 0.
💡 Why this works: The derivative d/dx[sign(x)] is zero for all x except at x = 0, where it is undefined due to the discontinuity. This is the correct final result based on the structure of the sign function.
[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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