If \(f(x) = x^3 - x\), find \(f'(x)\).

If \(f(x) = x^3 - x\), find \(f'(x)\).

• A. 2x + 1
• B. 3x^2 + 1
• C. 3x^2 - 1
• D. x^2 - 2

Answer: (C)

To find the derivative of the function \( f(x) = x^3 - x \), we apply the power rule of differentiation, which states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \). First, we differentiate each term in the function separately: 1. For the term \( x^3 \), the derivative is: \[ 3 \cdot x^{3-1} = 3x^2 \] 2. For the term \( -x \), which can be viewed as \( -1 \cdot x^1 \), the derivative is: \[ -1 \cdot 1 \cdot x^{1-1} = -1 \] Now, we combine the derivatives of both terms to find \( f'(x) \): \[ f'(x) = 3x^2 - 1 \] This matches the identified choice. The correct answer is derived directly from applying foundational rules of calculus to differentiate the provided cubic polynomial correctly, ensuring that we account for each term's exponent appropriately. Understanding this process helps reinforce the concepts of derivatives, particularly for polynomial functions, allowing

To find the derivative of the function ( f(x) = x^3 - x ), we apply the power rule of differentiation, which states that the derivative of ( x^n ) is ( n \cdot x^{n-1} ).

First, we differentiate each term in the function separately:

1. For the term ( x^3 ), the derivative is:

[

3 \cdot x^{3-1} = 3x^2

]

2. For the term ( -x ), which can be viewed as ( -1 \cdot x^1 ), the derivative is:

[

-1 \cdot 1 \cdot x^{1-1} = -1

]

Now, we combine the derivatives of both terms to find ( f'(x) ):

[

f'(x) = 3x^2 - 1

]

This matches the identified choice. The correct answer is derived directly from applying foundational rules of calculus to differentiate the provided cubic polynomial correctly, ensuring that we account for each term's exponent appropriately.

Understanding this process helps reinforce the concepts of derivatives, particularly for polynomial functions, allowing