Explain what is the rule applied to determine f'(x) if f(x)=(x^2+2)^4?

The rule applied to determine the first derivative of a
composed function is called "chain rule".

f'(x)
=[(x^2+2)^4]'*(x^2+2)'

First, we'll differentiate with
respect to x,the first factor of the product. We'll differentiate using power
rule.

[(x^2+2)^4]' =
4(x^2+2)^3

Now, we'll differentiate with respect to x, the
second factor:

(x^2+2)' = 2x +
0

(x^2+2)' =
2x

The first derivative of the given function
was determined using chain rule and it is: f'(x) =
8x*(x^2+2)^3.