Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)

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Question: Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)

Options:

Correct Answer: 4x^3 + 6x^2 - 1

Exam Year: 2023

Solution:

Using the power rule, f\'(x) = 4x^3 + 6x^2 - 1.

Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)

Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)

Step 1: Identify the function f(x) = x^4 + 2x^3 - x + 1.
Step 2: Recall the power rule for derivatives: if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Apply the power rule to each term in the function.
Step 4: For the first term x^4, the derivative is 4*x^(4-1) = 4x^3.
Step 5: For the second term 2x^3, the derivative is 2*3*x^(3-1) = 6x^2.
Step 6: For the third term -x, which is -1*x^1, the derivative is -1*1*x^(1-1) = -1.
Step 7: For the constant term +1, the derivative is 0 because the derivative of a constant is always 0.
Step 8: Combine all the derivatives from the previous steps: f'(x) = 4x^3 + 6x^2 - 1.

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.