Learn how to find the derivative of x^3 + x. The derivative is d/dx[x^3 + x].
Below is the graph of f(x) = x^3 + 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^3 + x is a sum of two terms, each of which involves a power of x. The Power Rule applies to both terms: x^3 and x.
💡 Why this works: The Power Rule is applicable because each term involves a power of x, making it possible to differentiate them individually by multiplying the exponent by the coefficient and subtracting one from the exponent.
Differentiate each term of x^3 + x using the Power Rule. For x^3, the derivative is 3x^2. For x, the derivative is 1.
💡 Why this works: Applying the Power Rule to x^3 gives 3x^2, and applying it to x gives 1. Thus, the derivative of x^3 + x is 3x^2 + 1.
Combine the results of the individual differentiations. The derivative of x^3 + x simplifies to 3x^2 + 1.
💡 Why this works: The expression for the derivative is already in its simplest form. Verifying it involves checking that the differentiation was done correctly for both terms.
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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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