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The problem is asking to find derivatives of two functions:

1. \( p'(9) \) where \( p(x) = f(x)g(x) \)
2. \( q'(4) \) where \( q(x) = \frac{f(x)}{g(x)} \)

The graph provided shows both \( f(x) \) and \( g(x) \), so let's use the product and quotient rules for differentiation.

### (a) \( p'(9) \) – Product Rule:
For \( p(x) = f(x)g(x) \), the derivative is given by the product rule:
\[
p'(x) = f'(x)g(x) + f(x)g'(x)
\]
To calculate \( p'(9) \), we need:
- \( f(9) \), \( f'(9) \)
- \( g(9) \), \( g'(9) \)

From the graph at \( x = 9 \):
- \( f(9) = 1 \)
- \( g(9) = 4 \)

Now, the slopes at \( x = 9 \) (to find the derivatives):
- \( f'(9) \): The graph of \( f(x) \) looks flat at \( x = 9 \), so \( f'(9) = 0 \).
- \( g'(9) \): From the graph, the slope of \( g(x) \) at \( x = 9 \) appears to be \( 1 \).

Now substitute into the product rule:
\[
p'(9) = f'(9)g(9) + f(9)g'(9) = 0 \times 4 + 1 \times 1 = 1
\]

So, \( p'(9) = 1 \).

### (b) \( q'(4) \) – Quotient Rule:
For \( q(x) = \frac{f(x)}{g(x)} \), the derivative is given by the quotient rule:
\[
q'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}
\]
To calculate \( q'(4) \), we need:
- \( f(4) \), \( f'(4) \)
- \( g(4) \), \( g'(4) \)

From the graph at \( x = 4 \):
- \( f(4) = 6 \)
- \( g(4) = 4 \)

Now, the slopes at \( x = 4 \):
- \( f'(4) = -1 \) (decreasing slope)
- \( g'(4) = -\frac{1}{2} \) (slightly decreasing slope)

Now substitute into the quotient rule:
\[
q'(4) = \frac{f'(4)g(4) - f(4)g'(4)}{g(4)^2} = \frac{(-1)(4) - (6)(-\frac{1}{2})}{4^2}
\]
\[
q'(4) = \frac{-4 + 3}{16} = \frac{-1}{16}
\]

So, \( q'(4) = -\frac{1}{16} \).

Would you like any further details or explanations?

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Here are some related questions:
1. What is the general product rule for differentiation?
2. How is the quotient rule derived from the product rule?
3. How do you estimate the slope from a graph visually?
4. What is the importance of checking points and slopes in real-world contexts?
5. How can these differentiation techniques be applied in physics or economics?

**Tip**: Always ensure your slope estimates from graphs are as precise as possible by looking at intervals between points.

Use the graph of f and g to find the derivatives of the following functions: (a) p'(9) where p(x) = f(x)g(x) and (b) q'(4) where q(x) = f(x)/g(x).

Learn how to find the derivatives p'(9) and q'(4) using the product and quotient rules from the given graph of functions f and g. This problem covers key calculus concepts including differentiation using the product and quotient rules, with a focus on interpreting graphs to find function values and slopes.

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