Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)

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Question: Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)

Options:

Correct Answer: 5x^4 - 9x^2 + 2

Exam Year: 2023

Solution:

Using the power rule, f\'(x) = 5x^4 - 9x^2 + 2.

Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)

Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)

Step 1: Identify the function f(x) = x^5 - 3x^3 + 2x.
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 the first term x^5: The derivative is 5*x^(5-1) = 5x^4.
Step 4: Apply the power rule to the second term -3x^3: The derivative is -3*3*x^(3-1) = -9x^2.
Step 5: Apply the power rule to the third term 2x: The derivative is 2*1*x^(1-1) = 2.
Step 6: Combine all the derivatives: f'(x) = 5x^4 - 9x^2 + 2.

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