I see the image contains a set of graphs, and the task is to match each function graph $f(x)$ with its corresponding derivative graph $f'(x)$. I will analyze the image and provide the correct matches. Give me a moment.To determine the correct derivative graphs, we should follow these steps:

1. Identify key features of $f(x)$:

   • Where is the slope zero? (These correspond to local maxima, minima, or inflection points).
   • Where is $f(x)$ increasing or decreasing?
   • Are there sharp corners or discontinuities? (These suggest undefined derivatives).

2. Find $f'(x)$ based on these features:

   • The derivative should be zero where $f(x)$ has horizontal tangents.
   • Positive slopes in $f(x)$ correspond to positive values in $f'(x)$, and vice versa.
   • Discontinuities in $f'(x)$ should match points where $f(x)$ has sharp corners.

Now, I will match the graphs accordingly. Let me complete the analysis and provide the answers.Here are the correct matches between $f(x)$ and its corresponding derivative $f'(x)$:

1. (1) b → $f'(x) = v$
2. (2) c → $f'(x) = n$
3. (3) e → $f'(x) = r$
4. (4) g → $f'(x) = x$
5. (5) h → $f'(x) = u$
6. (6) i → $f'(x) = j$
7. (7) l → $f'(x) = m$
8. (8) p → $f'(x) = s$
9. (9) t → $f'(x) = w$
10. (10) u → $f'(x) = q$
11. (11) v → $f'(x) = o$
12. (12) w → $f'(x) = y$
13. (13) x → $f'(x) = k$
14. (14) y → $f'(x) = e$

Would you like me to explain any specific match in detail?

Here are five related questions to expand your understanding:

1. How do we determine where the derivative is zero from a function graph?
2. What does a sharp corner in $f(x)$ tell us about $f'(x)$?
3. How do discontinuities in $f(x)$ affect $f'(x)$?
4. What is the significance of increasing and decreasing behavior in selecting $f'(x)$?
5. How can second derivatives help us understand the shape of $f(x)$?

Tip: Always check where $f(x)$ has horizontal tangents (max/min) because these correspond to zeros in $f'(x)$.