College Math Placement Practice Test

What is the derivative of f(x) = x^3?

1. 2x^2

2. 3x^2

3. x^2

4. 2x

The correct answer is: 3x^2

The derivative of a function gives the rate at which the function's output changes with respect to its input. For the function \( f(x) = x^3 \), we can apply the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is calculated as \( f'(x) = n \cdot x^{n-1} \). In this case, applying the power rule to \( f(x) = x^3 \): 1. Identify \( n \): Here, \( n = 3 \). 2. Apply the power rule: \( f'(x) = 3 \cdot x^{3-1} = 3 \cdot x^2 \). Thus, the derivative of \( f(x) = x^3 \) is \( 3x^2 \). This is why the correct answer is \( 3x^2 \). The other options do not align with the application of the power rule for this specific exponent.