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) ?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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