find the derivative of the function $$ f\left(x\right)=\frac{1}{3-2x}\text{ } $$at the point (-1, 1/5)

To find the derivative of the function

\[
f(x) = \frac{1}{3 - 2x}
\]

at the point $(-1, \frac{1}{5})$, we can follow these steps:

### Step 1: Differentiate the function
Using the chain rule, the derivative of $f(x)$ can be found. First, rewrite the function as:
\[
f(x) = (3 - 2x)^{-1}
\]

Now, apply the chain rule:

\[
f'(x) = -1 \cdot (3 - 2x)^{-2} \cdot (-2) = \frac{2}{(3 - 2x)^2}
\]

### Step 2: Evaluate the derivative at $x = -1$
Now substitute $x = -1$ into the derivative formula:

\[
f'(-1) = \frac{2}{(3 - 2(-1))^2} = \frac{2}{(3 + 2)^2} = \frac{2}{5^2} = \frac{2}{25}
\]

### Conclusion:
The derivative of the function at the point $(-1, \frac{1}{5})$ is:

\[
f'(-1) = \frac{2}{25}
\]

Let me know if you'd like further details or clarifications.

Here are 5 related questions:
1. How would the derivative change if the denominator was different, e.g., $3 + 2x$?
2. What is the second derivative of $f(x)$?
3. How do you interpret the value of the derivative geometrically at the point $(-1, \frac{1}{5})$?
4. What happens to the derivative as $x$ approaches infinity?
5. How does the chain rule apply to more complex functions?

**Tip:** When using the chain rule, remember that you differentiate the outer function and multiply by the derivative of the inner function.

Find the derivative of the function f(x) = 1/(3 - 2x) at the point (-1, 1/5).

This solution explains how to find the derivative of the function f(x) = 1/(3 - 2x) at the point (-1, 1/5) using the chain rule. Learn the step-by-step process for differentiation and evaluate the derivative at a specific point. Suitable for advanced high school or early college-level calculus students.

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