Learn how to find the derivative of x^2 + x + 1. The derivative is d/dx[x^2 + x + 1].
Below is the graph of f(x) = x^2 + 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.
The function f(x) = x^2 + x + 1 is a polynomial. The Power Rule applies to each term of the polynomial individually. The first term, x^2, is a power of x, so the Power Rule will be used to differentiate it. The second term, x, is a linear term, and the third term, 1, is a constant.
💡 Why this works: The structure of f(x) requires the Power Rule because each term is a simple polynomial expression. The Power Rule is designed to differentiate terms of the form x^n, where n is a constant exponent.
To apply the Power Rule to f(x) = x^2 + x + 1, differentiate each term separately. For x^2, apply the Power Rule: bring down the exponent (2), and decrease the exponent by 1, resulting in 2x. For x, the derivative is simply 1, because the exponent of x is 1. The derivative of the constant 1 is 0.
💡 Why this works: By applying the Power Rule to each term, we differentiate f(x) = x^2 + x + 1 to get f'(x) = 2x + 1. The derivative of the constant term is zero because constants do not change with x.
After applying the Power Rule to each term, the derivative is f'(x) = 2x + 1. This is the final simplified form of the derivative.
💡 Why this works: The result, f'(x) = 2x + 1, is the correct derivative because each term was differentiated correctly using the Power Rule. The derivative tells us the rate of change of the function at any value of x.
[Teaching explanation - to be filled]
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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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