Given the function f(x) = x4 , and suppose g(x) is some other function of x.

(a) Then the function F(x) = (g(x))^4 is the same thing as ...
a. f '(g(x)) · g'(x)
b. f(g(x))
c. f '(x)
d. f '(g(x))
e. g(f(x))

(b) By either the extended power or the chain rule, we have F '(x) =
a. g(4x^3)
b. 4 (g(x))^3 · g '(x)
c. 4 (g(x))^3
d. x^4 g'(x) + 4x^3g(x)
e. 4x^3 · g '(x)

(c) It follows that F '(2) =
a. 32(g '(x))
b. 4 · g(8) · g '(2)
c. 4 · (g(2))^3
d. g(32)
e. 4(g(2))^3 · g '(2)

Thanks ya'll!

To solve these questions, we need to use the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

(a) The function F(x) is defined as (g(x))^4. To find its derivative, we can use the chain rule. The derivative of g(x) with respect to x is g'(x). Applying the chain rule, we get:

F'(x) = 4(g(x))^3 * g'(x)

So the correct answer is option b: f(g(x)).

(b) To find F'(x), we apply the chain rule again. The derivative of (g(x))^4 with respect to x is:

F'(x) = 4(g(x))^3 * g'(x)

So the correct answer is option e: 4x^3 * g '(x).

(c) To evaluate F'(2), we substitute x = 2 into the expression obtained in (b). This gives us:

F'(2) = 4(2)^3 * g '(2)

So the correct answer is option e: 4(g(2))^3 * g '(2).

To summarize:
(a) The function F(x) = (g(x))^4 is the same thing as f(g(x)), which is option b.
(b) The derivative F'(x) = 4x^3 * g '(x), which is option e.
(c) F'(2) = 4(g(2))^3 * g '(2), which is option e.

Hope this helps! Let me know if you have any further questions.