Final answer:
To calculate y'(2) for the function given, apply the product rule for differentiation, substitute x = 2 into the derived expression, and simplify to find the value is 102.
Step-by-step explanation:
To find y'(2), which represents the derivative of the function y at x = 2, we first need to take the derivative of y with respect to x. The function is y = (x2 + x + 1)(x3 − x − 1). This product of functions suggests we should apply the product rule, which states that if u(x) and v(x) are functions of x, then the derivative u(x)v(x) is u'(x)v(x) + u(x)v'(x). After differentiating and applying the product rule:
The derivative y' is (2x + 1)(x3 - x - 1) + (x2 + x + 1)(3x2 - 1).
Next, we substitute x = 2 into y' and simplify to find y'(2):
y'(2) = (2(2) + 1)(23 - 2 - 1) + (22 + 2 + 1)(3(2)2 - 1)
= (5)(5) + (7)(11)
= 25 + 77
= 102.
So the value of y'(2) is 102.