Final answer:
The derivative of y = (x^2 + 1)(x^2 + x + 4) is y' = 4x^3 + 4x^2 + 10x + 1, found by applying the product rule and then simplifying the expression.
Step-by-step explanation:
The derivative of y = (x^2 + 1)(x^2 + x + 4) can be found using the product rule in calculus. The product rule states that if you have a function y = u(x)v(x), then the derivative y' is u'(x)v(x) + u(x)v'(x). In this case, u(x) = x^2 + 1 and v(x) = x^2 + x + 4.
First, we find the derivatives of u(x) and v(x):
u'(x) = 2x
v'(x) = 2x + 1
Next, we apply the product rule:
y' = u'(x)v(x) + u(x)v'(x)
Plugging in the derivatives we found:
y' = (2x)(x^2 + x + 4) + (x^2 + 1)(2x + 1)
Simplifying, we get:
y' = 2x^3 + 2x^2 + 8x + 2x^2 + 2x + 2x^3 + 1
Combining like terms:
y' = 4x^3 + 4x^2 + 10x + 1