https://file.aiquickdraw.com/mathbot/file/1726631179023o7iafn5w.jpg

The problem asks to find the derivative of \( f(g(x)) \) at \( x = 4 \) using the chain rule. Let's break it down step by step using the provided table.

### Step 1: Recall the Chain Rule
The derivative of \( f(g(x)) \) with respect to \( x \) is given by:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]

### Step 2: Use the Given Values from the Table
At \( x = 4 \):
- \( g(4) = 3 \)
- \( f'(g(4)) = f'(3) = 16 \)
- \( g'(4) = -4 \)

### Step 3: Apply the Chain Rule
Now substitute the values into the chain rule expression:
\[
\frac{d}{dx}[f(g(x))] \bigg|_{x=4} = f'(g(4)) \cdot g'(4) = f'(3) \cdot g'(4) = 16 \cdot (-4) = -64
\]

### Step 4: Conclusion
It seems there was an error in interpreting the values or rules in the image. Let me know if you want a closer breakdown.

Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative of f(g(x)) at x = 4 using the provided table.

Solve the problem using the chain rule to find the derivative of f(g(x)) at x = 4. The table provides values for f, g, and their derivatives at specific points. Learn how to apply the chain rule step-by-step for this calculus problem, suitable for advanced high school students in Grades 11-12.

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