Final answer:
The derivative of the function y = x³ - 8x² + 16x + 7 is found by applying the power rule for differentiation, resulting in 3x² - 16x + 16.
Step-by-step explanation:
The student has asked to find the derivative of the function y = x³ - 8x² + 16x + 7. To find the derivative, apply the power rule for differentiation, which states that the derivative of x to the power of n is n times x to the power of (n-1).
Performing the differentiation step-by-step:
- For the term x³, the derivative is 3x².
- For the term -8x², the derivative is -16x.
- For the term 16x, the derivative is 16, since the derivative of x is 1.
- The last term is a constant, 7, and the derivative of a constant is 0.
Therefore, the derivative of the function y with respect to x is 3x² - 16x + 16.