1. **Problem statement:** Find the first and second derivatives of the function $y = 3x^4 - 5x^2 + 2$. 2. **Formula and rules:** - The derivative of $x^n$ is $nx^{n-1}$. - The derivative of a constant is 0. - The derivative of a sum is the sum of the derivatives. 3. **First derivative calculation:** $$y' = \frac{d}{dx}(3x^4) - \frac{d}{dx}(5x^2) + \frac{d}{dx}(2) = 3 \cdot 4x^{4-1} - 5 \cdot 2x^{2-1} + 0 = 12x^3 - 10x$$ 4. **Second derivative calculation:** $$y'' = \frac{d}{dx}(12x^3) - \frac{d}{dx}(10x) = 12 \cdot 3x^{3-1} - 10 \cdot 1x^{1-1} = 36x^2 - 10$$ 5. **Explanation:** - We apply the power rule to each term. - Constants multiply the derivative of the variable part. - The second derivative is the derivative of the first derivative. **Final answers:** $$y' = 12x^3 - 10x$$ $$y'' = 36x^2 - 10$$