Find the derivatives of the functions in Exercises 1–42.


𝔂 = x³ - 3 (x² + π²)

Verified step by step guidance

1. Identify the function for which you need to find the derivative: \( y = x^3 - 3(x^2 + \pi^2) \).
2. Apply the power rule to differentiate \( x^3 \). The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Therefore, the derivative of \( x^3 \) is \( 3x^2 \).
3. Differentiate the term \( -3(x^2 + \pi^2) \). First, apply the constant multiple rule, which allows you to take the constant \(-3\) outside the differentiation. Then, differentiate \( x^2 \) using the power rule, which gives \( 2x \).
4. Since \( \pi^2 \) is a constant, its derivative is 0. Therefore, the derivative of \( x^2 + \pi^2 \) is \( 2x \).
5. Combine the results: The derivative of the function \( y = x^3 - 3(x^2 + \pi^2) \) is \( 3x^2 - 3(2x) \). Simplify this expression to obtain the final derivative.