To find the value of $(f \circ g)'(x)$ at $x = 49$, we will apply the chain rule.

The chain rule for the derivative of a composite function $(f \circ g)(x)$ is:

$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$

Given:

• $f(u) = f(u)$ (in terms of $u$)
• $g(x) = \sqrt{x}$
• $x = 49$

Now, let's break down the problem:

1. Find $g'(x)$: Since $g(x) = \sqrt{x}$, we differentiate $g(x)$:

   $g'(x) = \frac{d}{dx} \left( \sqrt{x} \right) = \frac{1}{2\sqrt{x}}$

2. Evaluate $g'(x)$ at $x = 49$:

   $g'(49) = \frac{1}{2\sqrt{49}} = \frac{1}{2 \times 7} = \frac{1}{14}$

3. Now, find $f'(u)$: We need to know the form of $f(u)$. If $f(u)$ is specified, we can find $f'(u)$. Otherwise, we cannot continue without knowing the specific form of $f(u)$. Could you provide the explicit function for $f(u)$?

Would you like to proceed with an assumption for $f(u)$ or provide the function $f(u)$? Let me know!

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Further questions:

1. How does the chain rule work in more complex composite functions?
2. What happens if the inner function $g(x)$ is more complicated than $\sqrt{x}$?
3. How can we apply the chain rule when there are multiple functions composed together?
4. How do you find the second derivative of composite functions?
5. How does implicit differentiation relate to the chain rule?

Tip: Always check if the chain rule is the best approach when dealing with composite functions.