https://file.aiquickdraw.com/mathbot/file/1726718885356qc7mc643.jpg

Let's break down the problem step by step.

### Part (a)
You are asked to simplify the following difference quotient:

\[
\frac{s(t+h) - s(t)}{h}
\]

where \( s(t) = \frac{t + 1}{7 - t} \).

#### Step-by-step process for simplification:

1. Substitute \( s(t) = \frac{t + 1}{7 - t} \) and \( s(t+h) = \frac{(t+h) + 1}{7 - (t + h)} \).
   \[
   s(t+h) = \frac{t + h + 1}{7 - t - h}
   \]
   \[
   s(t) = \frac{t + 1}{7 - t}
   \]

2. Form the difference quotient:
   \[
   \frac{s(t+h) - s(t)}{h} = \frac{\frac{t + h + 1}{7 - t - h} - \frac{t + 1}{7 - t}}{h}
   \]

3. To simplify this, you will need to combine the two fractions by finding a common denominator. The common denominator is \( (7 - t - h)(7 - t) \).

4. Simplify the numerator:
   \[
   \frac{(t + h + 1)(7 - t) - (t + 1)(7 - t - h)}{(7 - t - h)(7 - t)}
   \]

5. This can be expanded, simplified, and then divided by \( h \).

### Part (b)
This asks you to calculate the instantaneous velocity, which is the derivative of \( s(t) \), i.e., \( v(t) = s'(t) \).

Use the limit definition of the derivative:
\[
v(t) = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h}
\]

From part (a), you can compute the derivative using the simplified form of the difference quotient.

### Part (c)
This asks for the object's velocity at \( t = 6 \). Once you have the derivative from part (b), plug in \( t = 6 \) to find the velocity.

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Let me know if you'd like more details or help with any part of the steps. Here are 5 related questions you can explore:

1. How do you simplify complex rational expressions?
2. What is the process for finding derivatives using the difference quotient?
3. How do you interpret the velocity function in terms of motion?
4. What role does the limit play in defining instantaneous velocity?
5. How do you apply the chain rule when differentiating complex functions?

**Tip:** Always ensure to simplify expressions step by step to avoid mistakes when applying the difference quotient.

Simplify the difference quotient s(t+h) - s(t) / h where s(t) = (t + 1) / (7 - t). Use the result to calculate the instantaneous velocity at t = 6 seconds.

Learn how to simplify the difference quotient for s(t) = (t + 1) / (7 - t) and use it to find the instantaneous velocity at t = 6 seconds. This problem covers differentiation and limit concepts, suitable for college-level calculus students.

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