What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)

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Question: What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)

Options:

Correct Answer: 6x + 5

Exam Year: 2021

Solution:

The derivative f\'(x) = d/dx(3x^2 + 5x - 7) = 6x + 5.

What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)

What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)

Step 1: Identify the function we want to differentiate, which is f(x) = 3x^2 + 5x - 7.
Step 2: Recall the power rule for derivatives: if f(x) = ax^n, then f'(x) = n * ax^(n-1).
Step 3: Apply the power rule to the first term, 3x^2. Here, a = 3 and n = 2. So, the derivative is 2 * 3 * x^(2-1) = 6x.
Step 4: Apply the power rule to the second term, 5x. Here, a = 5 and n = 1. So, the derivative is 1 * 5 * x^(1-1) = 5.
Step 5: The last term is -7, which is a constant. The derivative of a constant is 0.
Step 6: Combine the derivatives from Steps 3, 4, and 5. So, f'(x) = 6x + 5 + 0.
Step 7: Simplify the expression. The final derivative is f'(x) = 6x + 5.

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