Final answer:
To find the derivative y' (also known as f'(x)) for the function y=(x³)/(3) + (x³)/(2)−2x+2, combine like terms and apply the power rule and the constant multiple rule to obtain f'(x) = (5/2)x² - 2.
Step-by-step explanation:
To find the derivative y' of the function y=(x³)/(3) + (x³)/(2)−2x+2, also known as f'(x), we will apply basic differentiation rules. These include the power rule, the constant multiple rule, and the sum rule. The power rule states that the derivative of xn with respect to x is n*xn-1. The constant multiple rule allows us to take constants out of the derivative, and the sum rule allows us to differentiate terms separately.
To apply these rules to the given function:
- First, combine like terms, so y = (5/6)x³ - 2x + 2.
- Apply the power rule to differentiate (5/6)x³, which gives us (5/6)*3*x2 = (5/2)x².
- Use the power rule to differentiate -2x, which gives us -2.
- The derivative of the constant +2 is 0 since the derivative of any constant is 0.
Putting it all together, the derivative of y with respect to x (y' or f'(x)) is: f'(x) = (5/2)x² - 2.