Find the derivative of f(x) = 3x^2 + 5x - 7.

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Question: Find the derivative of f(x) = 3x^2 + 5x - 7.

Options:

Correct Answer: 6x + 5

Solution:

Using the power rule, f\'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.

Find the derivative of f(x) = 3x^2 + 5x - 7.

Find the derivative of f(x) = 3x^2 + 5x - 7.

Correct Answer: 6x + 5

Step 1: Identify the function f(x) = 3x^2 + 5x - 7.
Step 2: Recognize that we need to find the derivative, which tells us the rate of change of the function.
Step 3: Apply the power rule for derivatives: If f(x) = ax^n, then f'(x) = n * ax^(n-1).
Step 4: For the first term 3x^2, apply the power rule: The derivative is 2 * 3 * x^(2-1) = 6x.
Step 5: For the second term 5x, apply the power rule: The derivative is 1 * 5 * x^(1-1) = 5.
Step 6: For the constant term -7, the derivative is 0 because the derivative of any constant is 0.
Step 7: Combine the derivatives from Steps 4, 5, and 6: f'(x) = 6x + 5 + 0.
Step 8: Simplify the expression: f'(x) = 6x + 5.