Learn how to find the derivative of x^4 + x^2. The derivative is d/dx[x^4 + x^2].
Below is the graph of f(x) = x^4 + x^2 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 x^4 + x^2 is made up of two terms that are powers of x, which is why we use the Power Rule for differentiation. Each term involves a simple monomial of the form x^n, where n is a constant exponent.
💡 Why this works: The Power Rule is ideal for this function because each term fits the form of a power function, and the rule applies directly to terms like x^4 and x^2.
To differentiate x^4 + x^2, we apply the Power Rule to each term separately. For x^4, the derivative is 4x^3, and for x^2, the derivative is 2x.
💡 Why this works: The Power Rule tells us to multiply the exponent by the coefficient (which is 1 in both terms) and reduce the exponent by one. Therefore, d/dx[x^4] = 4x^3 and d/dx[x^2] = 2x.
After applying the Power Rule to both terms, we combine the results to get the derivative d/dx[x^4 + x^2] = 4x^3 + 2x.
💡 Why this works: This is the correct final form of the derivative, as we have accurately applied the Power Rule to both terms and simplified the expression.
[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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