The given problem is:

$F(x) = \sqrt{g(x^2)}, \quad g(4) = 4, \quad g'(4) = 3$

We are tasked with finding $F'(2)$.

Step 1: Apply the Chain Rule

To differentiate $F(x) = \sqrt{g(x^2)}$, we use the chain rule. Rewriting $\sqrt{g(x^2)}$ as $[g(x^2)]^{1/2}$, the derivative of $F(x)$ is:

[ F'(x) = \frac{1}{2}[g(x^2)]^{-1/2} \cdot \frac{d}{dx}[g(x^2)] ]

Now, apply the chain rule to $g(x^2)$. The derivative of $g(x^2)$ with respect to $x$ is:

$\frac{d}{dx}[g(x^2)] = g'(x^2) \cdot 2x$

Thus, the derivative of $F(x)$ becomes:

$F'(x) = \frac{1}{2}[g(x^2)]^{-1/2} \cdot g'(x^2) \cdot 2x$

Simplifying:

$F'(x) = \frac{x \cdot g'(x^2)}{\sqrt{g(x^2)}}$

Step 2: Substitute $x = 2$

Now, let's find $F'(2)$:

$F'(2) = \frac{2 \cdot g'(2^2)}{\sqrt{g(2^2)}}$

We know that $g(4) = 4$ and $g'(4) = 3$, so substitute these values:

$F'(2) = \frac{2 \cdot 3}{\sqrt{4}} = \frac{2 \cdot 3}{2} = 3$

Thus, $F'(2) = 3$.

Final Answer:

$F'(2) = 3$

Would you like further clarification or additional details?

Here are 5 related questions:

1. How is the chain rule applied to composite functions like $F(x)$?
2. What happens when you differentiate a function involving a square root?
3. How would the problem change if $g(x^2)$ were a different type of function?
4. Can the chain rule be applied to multiple nested functions?
5. What are some other scenarios where $g'(x)$ is needed for derivative calculations?

Tip: When dealing with composite functions like $g(x^2)$, it's essential to recognize how to apply the chain rule step by step.