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Given that f'(x) = √(3x + 6) and g(x) = x² - 3, find F'(x) if F(x) = f(g(x)).

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Final answer:

To find F'(x) if F(x) = f(g(x)), first find the derivative of g(x) and the derivative of f(x). Then substitute g(x) into f'(x) using the chain rule. Finally, simplify the expression to get F'(x).

Step-by-step explanation:

To find F'(x) if F(x) = f(g(x)), we need to use the chain rule of differentiation. The chain rule states that the derivative of a composition of two functions is the derivative of the outer function multiplied by the derivative of the inner function.

First, we find the derivative of g(x): g'(x) = 2x. Then, we find the derivative of f(x): f'(x) = √(3x + 6).

Finally, we substitute g(x) into f'(x): F'(x) = f'(g(x)) * g'(x).

Therefore, F'(x) = √(3(g(x)) + 6) * 2x.

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