Calculus of a Single Variable, Chapter 2, 2.3, Section 2.3, Problem 14

You need to evaluate first y', using the product rule, such that:
y' = (x^2 - 3x + 2)'(x^3 + 1) + (x^2 - 3x + 2)(x^3 + 1)'
y' = (2x - 3)(x^3 + 1) + (x^2 - 3x + 2)(3x^2)
y' = 2x^4 + 2x - 3x^3 - 3 + 3x^4 - 9x^3 + 6x^2
Combining like terms yields:
y' = 5x^4 - 12x^3 + 6x^2 + 2x - 3
You may evaluate now f'(c) at c = 2, replacing 2 for x in equation of f'(x):
y' = 5*2^4 - 12*2^3 + 6*2^2 + 2*2 - 3
y' = 80 - 96 + 24 + 4 - 3
y' = 105 - 96 => y' = 9
Hence, evaluating the derivative of the function yields y' = 5x^4 - 12x^3 + 6x^2 + 2x - 3 and evaluating the value of derivative at c= 2, yields y' = 9.