Learn how to find the derivative of x^5 + 3x. The derivative is d/dx[x^5 + 3x].
Below is the graph of f(x) = x^5 + 3x 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^5 + 3x is a sum of two terms: x^5 and 3x. The Power Rule is applicable here because both terms involve powers of x. For x^5, the Power Rule applies directly. For 3x, the Power Rule still applies, but we treat it as x^1.
💡 Why this works: The function is made up of a polynomial where each term follows the rules of differentiation for powers of x. The Power Rule is the ideal tool to differentiate each term independently.
To find the derivative of x^5, we apply the Power Rule, which states that the derivative of x^n is n*x^(n-1). Applying this to x^5 gives us 5x^4. For 3x, we apply the Power Rule again, treating it as x^1. The derivative of x^1 is simply 1, so the derivative of 3x is 3.
💡 Why this works: By applying the Power Rule to each term, we differentiate x^5 to 5x^4 and 3x to 3. Combining these results gives the derivative d/dx[x^5 + 3x] = 5x^4 + 3.
The derivative of x^5 + 3x simplifies to 5x^4 + 3. This result is straightforward and reflects the sum of the individual derivatives of each term.
💡 Why this works: The derivative is correct because we applied the Power Rule properly to both terms. Each step follows logically, and the simplified form of the derivative is consistent with the rules of differentiation.
[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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