Find a formula for y′y′ and determine the slope y′||x=5y′|x=5 for the following function.

y=e^x/−7e^x−6

To find the formula for \( y' \) and determine the slope \( y'|_{x=5} \) for the function \( y = \frac{e^x}{-7e^x - 6} \), we will follow these steps:

1. **Differentiate the function \( y \) with respect to \( x \).**

Given:
\[ y = \frac{e^x}{-7e^x - 6} \]

We use the quotient rule for differentiation, which states:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]

Here:
\[ u = e^x \]
\[ v = -7e^x - 6 \]

First, find \( u' \) and \( v' \):
\[ u' = e^x \]
\[ v' = -7e^x \]

Now, apply the quotient rule:
\[ y' = \frac{u'v - uv'}{v^2} \]
\[ y' = \frac{e^x(-7e^x - 6) - e^x(-7e^x)}{(-7e^x - 6)^2} \]
\[ y' = \frac{-7e^{2x} - 6e^x + 7e^{2x}}{(-7e^x - 6)^2} \]
\[ y' = \frac{-6e^x}{(-7e^x - 6)^2} \]

So, the derivative of the function \( y \) is:
\[ y' = \frac{-6e^x}{(-7e^x - 6)^2} \]

2. **Evaluate the slope at \( x = 5 \):**

To find \( y' \) at \( x = 5 \), substitute \( x = 5 \) into the derivative:
\[ y'|_{x=5} = \frac{-6e^5}{(-7e^5 - 6)^2} \]

Simplify the expression:
\[ y'|_{x=5} = \frac{-6e^5}{( - 7e^5 - 6)^2} \]

This is the slope of the function at \( x = 5 \).

Would you like any further details or have any questions on this solution?

Here are 8 related questions that might interest you:

1. How do you apply the quotient rule to differentiate a function?
2. What is the significance of the quotient rule in calculus?
3. Can you find the second derivative of the function \( y = \frac{e^x}{-7e^x - 6} \)?
4. How would you determine the critical points of the function \( y = \frac{e^x}{-7e^x - 6} \)?
5. What is the behavior of the function \( y = \frac{e^x}{-7e^x - 6} \) as \( x \) approaches infinity?
6. How can you find the horizontal asymptotes of the function \( y = \frac{e^x}{-7e^x - 6} \)?
7. What are the steps to determine the concavity of the function \( y = \frac{e^x}{-7e^x - 6} \)?
8. How does the exponential function \( e^x \) affect the behavior of the function \( y = \frac{e^x}{-7e^x - 6} \)?

**Tip:** When differentiating functions, always ensure to simplify the expression wherever possible to make further calculations easier.

Learn how to find the derivative and determine the slope of the function y = e^x / (-7e^x - 6) at x = 5. This problem involves using the quotient rule for differentiation and evaluating exponential functions. Suitable for advanced high school and college students.

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