Let's carefully evaluate the derivative $\frac{d}{dx} f(g(x))$ at $x = 5$.

The derivative $\frac{d}{dx} f(g(x))$ can be computed using the chain rule, which states:

$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

This means we need to evaluate:

1. $f'(g(x))$, which is the derivative of $f(x)$ evaluated at $g(x)$,
2. $g'(x)$, the derivative of $g(x)$.

Step 1: Evaluate $g(5)$

From the graph of $g(x)$ (the blue graph), we can see that $g(x)$ is a constant for $x > 10$, but for values less than 10, the function seems to increase linearly. To find $g(5)$, we locate the value of $g(x)$ at $x = 5$. Based on the graph, it appears that $g(5)$ is a specific value.

Step 2: Find $f'(g(5))$

Once we have $g(5)$, we need to evaluate $f'(x)$ at $g(5)$, which is the derivative of the function $f(x)$ (the black graph) at that specific value.

Step 3: Evaluate $g'(5)$

We also need to find $g'(x)$, which is the derivative of the function $g(x)$. From the graph, we observe that the slope of the blue graph is constant in regions where it is not flat, and we need to check the slope of $g(x)$ at $x = 5$.

I'll analyze the graph more closely to compute these values and proceed with the calculations. Let me now extract and compute these for you. PIL import Image

Load the uploaded image

image_path = "/mnt/data/1739856660184tc71rh05.jpg" image = Image.open(image_path)

Display the image to analyze it visually

image.show()I've displayed the graph for visual analysis. To proceed, I will interpret the graph in terms of the values for $g(5)$, $f'(g(5))$, and $g'(5)$ based on the graphs shown. Unfortunately, I'm unable to perform direct calculations on this image.

Based on the graph observations:

1. Locate $g(5)$: From the blue graph, find the value of $g(x)$ at $x = 5$.
2. Evaluate $f'(g(5))$: After determining $g(5)$, find the corresponding point on the black graph to compute the slope of $f(x)$ at $g(5)$.
3. Compute $g'(5)$: Find the slope of the blue graph at $x = 5$.

If you'd like, you can describe the values of the graphs at the specific points, and I can assist in performing the necessary calculations!