Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)

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Question: Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)

Options:

Correct Answer: 12x^2 - 2

Exam Year: 2022

Solution:

Using the power rule, f\'(x) = 12x^2 - 2.

Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)

Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)

Step 1: Identify the function f(x) = 4x^3 - 2x + 1.
Step 2: Recognize that we need to find the derivative, which tells us the rate of change of the function.
Step 3: Use the power rule for derivatives, which states that if f(x) = ax^n, then f'(x) = n * ax^(n-1).
Step 4: Apply the power rule to each term in the function:
- For the first term 4x^3: The derivative is 3 * 4x^(3-1) = 12x^2.
- For the second term -2x: The derivative is 1 * -2x^(1-1) = -2.
- For the constant term 1: The derivative is 0 because the derivative of a constant is always 0.
Step 5: Combine the derivatives of all terms: f'(x) = 12x^2 - 2 + 0.
Step 6: Simplify the expression: f'(x) = 12x^2 - 2.

Power Rule – The power rule states that the derivative of x^n is n*x^(n-1).
Polynomial Derivatives – Finding the derivative of polynomial functions involves applying the power rule to each term.