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To solve the problem, we need to find the derivative \(\frac{dy}{dx}\) of the given function:

\[
y = \frac{1}{x - 5}
\]

### Step-by-Step Solution

1. **Rewrite the function using a negative exponent**:
   \[
   y = (x - 5)^{-1}
   \]
   This transformation allows us to avoid using the quotient rule.

2. **Use the General Power Rule for differentiation**:
   The general power rule is:
   \[
   \frac{d}{dx} \left( u^n \right) = n u^{n-1} \cdot \frac{du}{dx}
   \]
   Here, \( u = x - 5 \) and \( n = -1 \).

3. **Differentiate**:
   Applying the power rule:
   \[
   \frac{dy}{dx} = -1 \cdot (x - 5)^{-2} \cdot \frac{d}{dx}(x - 5)
   \]
   Since the derivative of \( x - 5 \) with respect to \( x \) is just 1:
   \[
   \frac{dy}{dx} = - (x - 5)^{-2}
   \]

4. **Final Answer**:
   \[
   \frac{dy}{dx} = - \frac{1}{(x - 5)^2}
   \]

Thus, the derivative is:

\[
\frac{dy}{dx} = - \frac{1}{(x - 5)^2}
\]

---

Would you like more details or have any questions? Here are some follow-up questions for further exploration:

1. How would you apply the quotient rule to verify the result?
2. Can this method be used for more complex rational functions?
3. What happens to the derivative if we shift the value of \( 5 \) to another number?
4. How can you graph the function \( y = \frac{1}{x - 5} \) and its derivative?
5. How would the second derivative of this function look?

**Tip**: Always check if a simpler rule, like the power rule, can replace more complex methods such as the quotient rule for faster calculations.

Find dy/dx without using the quotient rule; rewrite the function using a negative exponent and use the general power rule to find the derivative. Given y = 1 / (x - 5).

Learn how to find the derivative of y = 1 / (x - 5) without using the quotient rule. This step-by-step solution rewrites the function using a negative exponent and applies the power rule for differentiation. Suitable for students in Grades 11-12 or early college calculus.

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