Determine the derivative of f(x) = x^3 - 4x + 7. (2023)

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Question: Determine the derivative of f(x) = x^3 - 4x + 7. (2023)

Options:

Correct Answer: 3x^2 - 4

Exam Year: 2023

Solution:

Using the power rule, f\'(x) = 3x^2 - 4.

Determine the derivative of f(x) = x^3 - 4x + 7. (2023)

Determine the derivative of f(x) = x^3 - 4x + 7. (2023)

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

Power Rule – The power rule states that the derivative of x^n is n*x^(n-1), which is applied to each term of the polynomial.
Constant Rule – The derivative of a constant is zero, which is relevant for the constant term in the function.