composite function of (x^2 + 4x + 6)^5 ?

Question

first i have to identify the inner function u = g(x) and the outer function y = f(u).
then i need find the derivative dy/dx.
i already know the answer is 10(x^2 + 4x + 6)^4(x + 2).

I really want to understand this problem and im not being lazy or anything. any help is appreciated.

Answer 1

Primarily this is an expression raised to the fifth power, and so that is the outer function:
f(x) = x^5, or f(u) = u^5

Clearly, to get your function, you need to replace x by
x^2 + 4x + 6, and so that is the inner function, g(x).
Checking, note that
f(g(x)) = f(x^2 + 4x + 6)
= (x^2 + 4x + 6)^5

I expect you know the "chain rule" for differentiating a composite function:
dy/dx = (dy/du)*(du/dx)
= 5u^4*(2x + 4) and you finish this by replacing
u with x^2 + 4x + 6 and taking out the common factor from
2x + 4.

Answer 2

ok x^2+4x+6
and your function f(u) is
u^5

Answer 3

Looks like your g(x) is x^2+4x+6
and your outer function f(u) is
u^5
so you have f(g(x))
Use chain rule for derivative:

f'(x) = f'(g(x))*g'(x)