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     7 months ago  

    Step 1: Identify the function

    The given function is f(x) = 3x^3 - 2x^2 + x - 5.

    Step 2: Apply the power rule for differentiation

    To find the derivative of f(x), we'll apply the power rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

    Step 3: Differentiate each term

    Using the power rule, we'll differentiate each term in the function:

    • The derivative of 3x^3 is 3*3x^(3-1) = 9x^2.
    • The derivative of -2x^2 is -2*2x^(2-1) = -4x.
    • The derivative of x is 1x^(1-1) = 1.
    • The derivative of -5 is 0, since the derivative of a constant is always 0.

    Step 4: Combine the derivatives

    Now, we'll combine the derivatives of each term to find the derivative of the entire function:
    f'(x) = 9x^2 - 4x + 1.

    The final answer is: $\boxed{9x^2 - 4x + 1}$

    Note: If you have any questions or need further clarification, feel free to ask.