Learn how to find the derivative of x^2 + 3x. The derivative is d/dx[x^2 + 3x].
Below is the graph of f(x) = x^2 + 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^2 + 3x is a polynomial, and it requires the Power Rule for differentiation. This rule applies to terms where the variable x is raised to a power, such as x^2 and x^1 in the case of 3x.
💡 Why this works: The Power Rule is used for differentiating terms of the form x^n, where n is a constant. For each term, we multiply by the exponent and decrease the exponent by 1. This is the first key reason we apply the Power Rule to differentiate x^2 + 3x.
Now we apply the Power Rule to each term of x^2 + 3x. For the x^2 term, the derivative is 2x (since the power of x is 2). For the 3x term, the derivative is simply 3 (since the power of x is 1, and the derivative of x is 1).
💡 Why this works: The derivative of x^2 is 2x, and the derivative of 3x is 3. Therefore, the derivative of x^2 + 3x is f'(x) = 2x + 3.
After applying the Power Rule to each term, we have the derivative 2x + 3. There are no further simplifications necessary, and the result is in its simplest form.
💡 Why this works: The final derivative, f'(x) = 2x + 3, represents the slope of the tangent line to the curve at any point x. This confirms that the calculation is correct.
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[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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