Learn how to find the derivative of min(x,1). The derivative is d/dx[min(x,1)].
Below is the graph of f(x) = min(x,1) 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.
We first recognize that min(x,1) is a piecewise function. It behaves differently depending on whether x is less than, equal to, or greater than 1. For x < 1, the function is equivalent to f(x) = x, and for x ≥ 1, the function becomes f(x) = 1.
💡 Why this works: Since the Power Rule applies to functions of the form x^n (where n is a constant), we can apply the Power Rule to the piece of the function where x < 1 (since f(x) = x). For x ≥ 1, the function is constant, so the derivative of that part is zero.
We differentiate f(x) = min(x,1) by considering the two regions. For x < 1, the function is just x, so the derivative is 1. For x ≥ 1, the function is constant (f(x) = 1), so the derivative is 0.
💡 Why this works: Thus, the derivative of min(x,1) is piecewise: f'(x) = 1 for x < 1 and f'(x) = 0 for x ≥ 1.
We now have the piecewise derivative: f'(x) = 1 if x < 1 and f'(x) = 0 if x ≥ 1. This reflects the fact that the function is increasing when x is less than 1 and constant when x is greater than or equal to 1.
💡 Why this works: This result is correct because for x < 1, the function behaves like f(x) = x, whose derivative is 1, and for x ≥ 1, the function behaves like a constant (f(x) = 1), whose derivative is 0.
[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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