<|im_start|>user
What is the derivative of the function \( f(x) = x^3 + 2x^2 - 5x + 1 \)?<|im_end|>
<|im_start|>assistant
The derivative is \( f'(x) = 3x^2 + 4x - 5 \) because it correctly applies the power rule of differentiation. According to calculus, the power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). This means that for each term in the function, you multiply by the exponent and then decrease the exponent by one, which leads to the correct derivative.<|im_end|>

<|im_start|>user
What is the derivative of the function \( f(x) = x^3 + 2x^2 - 5x + 1 \)?<|im_end|>
<|im_start|>assistant
The derivative is \( f'(x) = 2x^2 + 3x - 5 \). This answer is incorrect because it misapplies the power rule. Specifically, it incorrectly assigns the coefficients and exponents to the function's terms, leading to an erroneous derivative calculation. Thus, the correct expression is essential for maintaining accuracy in calculus operations.<|im_end|>

What is the derivative of the function \( f(x) = x^3 + 2x^2 - 5x + 1 \)?