https://file.aiquickdraw.com/mathbot/file/1728168626872kprwh1hh.jpg

Let's break down the problem shown in the image.

We are given two functions \( f(x) \) and \( g(x) \), and we are tasked with finding:

(a) \( u'(1) \), where \( u(x) = f(x) g(x) \).

(b) \( v'(5) \), where \( v(x) = \frac{f(x)}{g(x)} \).

### Part (a): Finding \( u'(1) \)
For \( u(x) = f(x)g(x) \), we need to apply the **product rule**:
\[
u'(x) = f'(x)g(x) + f(x)g'(x)
\]
At \( x = 1 \), we need to find \( f(1) \), \( f'(1) \), \( g(1) \), and \( g'(1) \).

From the graph:
- \( f(1) = 0 \) (since the function \( f(x) \) crosses the x-axis at \( x = 1 \)),
- \( g(1) = 0 \) (the graph of \( g(x) \) also crosses the x-axis at \( x = 1 \)),
- \( f'(1) \), which is the slope of the line at \( x = 1 \), is negative (approx. \( -2 \)),
- \( g'(1) \) is positive (approx. \( 2 \)).

Using the product rule at \( x = 1 \):
\[
u'(1) = f'(1)g(1) + f(1)g'(1) = (-2)(0) + (0)(2) = 0
\]
So, \( u'(1) = 0 \), which is correctly marked in your solution.

### Part (b): Finding \( v'(5) \)
For \( v(x) = \frac{f(x)}{g(x)} \), we need to apply the **quotient rule**:
\[
v'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]
At \( x = 5 \), we need to find \( f(5) \), \( f'(5) \), \( g(5) \), and \( g'(5) \).

From the graph:
- \( f(5) = 1 \),
- \( g(5) = -2 \),
- \( f'(5) \) (the slope of \( f(x) \) at \( x = 5 \)) is approximately \( 0.5 \),
- \( g'(5) \) (the slope of \( g(x) \) at \( x = 5 \)) is \( 1 \).

Now, applying the quotient rule:
\[
v'(5) = \frac{(0.5)(-2) - (1)(1)}{(-2)^2} = \frac{-1 - 1}{4} = \frac{-2}{4} = -\frac{1}{2}
\]

Thus, \( v'(5) = -\frac{1}{2} \).

Let me know if you need more details, or have further questions! Here are some related questions for you:

1. What is the difference between the product rule and quotient rule?
2. How do you determine the slope of a linear function from a graph?
3. Can the derivative of a product ever be zero? Why or why not?
4. How does the sign of the slope affect the behavior of the derivative?
5. What does a zero derivative tell you about the function at that point?

**Tip:** Always verify the slopes carefully from the graph when calculating derivatives using product or quotient rules.

If f and g are functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x). Find u'(1) and v'(5).

Learn how to solve derivative problems using the product and quotient rules. This problem focuses on calculating u'(1) for u(x) = f(x)g(x) and v'(5) for v(x) = f(x)/g(x) based on the given graphs of f and g. Key calculus concepts are covered, including derivative formulas.

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