Functions f,g, and h are defined as follows: g(x)=f(x^2), f(x)=h(x^3+1) and h'(x)=2x+1. What is g'(x) ?

We notice that we have to determine the function g(x), in
order to calculate it's first derivative.

We also notice
that we'll have to detemine f(x) because g(x) =
f(x^2).

Since f(x) = h(x^3 + 1), we need to find out the
function h(x).

Since we know the expression of the
derivative of the funcion h(x), we'll evaluate the indefinite integral of h'(x) to
determine the primitive function h(x).

`int` h'(x) dx =
`int` (2x+1)dx

`int` (2x+1)dx = `int` 2x dx + `int`
dx

`int` (2x+1)dx = 2*x^2/2 + x +
C

`int` (2x+1)dx = x^2 + x +
C

Therefore h(x)  = x^2 + x

We
can determine f(x) substituting x by the expression x^3 + 1 in the expression of
h(x).

h(x^3 + 1) = (x^3 + 1)*(x^3 + 1 +
1)

h(x^3 + 1) = (x^3 + 1)*(x^3
+2)

Therefore, f(x) = h(x^3 + 1) = (x^3 + 1)*(x^3
+2).

Now, we can determine
g(x):

g(x) = f(x^2) = (x^6 + 1)*(x^6
+2)

We'll differentiate with respect to x and we'll
get:

g'(x) = [(x^6 + 1)*(x^6
+2)]'

We'll use the product
rule:

g'(x) = [(x^6 + 1)]'*(x^6 +2) + (x^6 + 1)*[(x^6
+2)]'

g'(x) = 6x^5*(x^6 +2) + 6x^5*(x^6
+1)

g'(x) = 6x^5*(x^6 + 2 + x^6 +
1)

g'(x) = 6x^5*(2x^6 +
3)

g'(x) = 12x^11 +
18x^5

The requested derivative of the
function g(x) is g'(x) = 12x^11 + 18x^5.