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To help you with this problem, let's break down the given question. You are asked to find the derivative of two functions based on their composition. 

The graphs of \( f(x) \) and \( g(x) \) are given. 

- \( h(x) = f(g(x)) \)
- \( s(x) = g(f(x)) \)

You need to compute the following derivatives:

### Part (a): Find \( h'(-1) \)
To compute \( h'(x) \), we apply the chain rule:

\[
h'(x) = f'(g(x)) \cdot g'(x)
\]

To find \( h'(-1) \), we need:
1. \( g(-1) \) (which can be read from the graph of \( g(x) \)),
2. \( f'(g(-1)) \), the slope of \( f(x) \) at \( g(-1) \),
3. \( g'(-1) \), the slope of \( g(x) \) at \( x = -1 \).

### Part (b): Find \( s'(1) \)
Similarly, for \( s(x) = g(f(x)) \), use the chain rule:

\[
s'(x) = g'(f(x)) \cdot f'(x)
\]

To find \( s'(1) \), we need:
1. \( f(1) \) (which can be read from the graph of \( f(x) \)),
2. \( g'(f(1)) \), the slope of \( g(x) \) at \( f(1) \),
3. \( f'(1) \), the slope of \( f(x) \) at \( x = 1 \).

### Let’s now proceed with evaluating these steps:
- Looking at the graph, determine the values of \( g(-1) \), \( f(g(-1)) \), and their respective slopes.
- Similarly, determine \( f(1) \), \( g'(f(1)) \), and their respective slopes.

Would you like me to walk through these calculations based on the graph?

The graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(f(x)). Find the derivative. (If an answer does not exist, enter DNE.) (a) Find h'(-1). (b) Find s'(1).

Learn how to solve a problem involving derivatives of composed functions using the chain rule. This detailed guide walks you through finding h'(-1) and s'(1) with the help of graph interpretations for the functions f(x) and g(x). Suitable for high school and early college students studying calculus.

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