find the derivative of the function $$ f\left(x\right)=\frac{1}{3-2x} $$ on the point (-1, 1/5) without using the chain rule

To find the derivative of the function

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

at the point $(-1, \frac{1}{5})$ **without using the chain rule**, we can rewrite the function and apply the quotient rule. Here's how:

### Step 1: Rewrite the function
We have the function:

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

This can be seen as a quotient:

\[
f(x) = \frac{u(x)}{v(x)}
\]

where $ u(x) = 1 $ (constant) and $ v(x) = 3 - 2x $.

### Step 2: Apply the quotient rule
The quotient rule states that for a function $ f(x) = \frac{u(x)}{v(x)} $, the derivative is:

\[
f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]

- $ u(x) = 1 $, so $ u'(x) = 0 $ (since the derivative of a constant is 0).
- $ v(x) = 3 - 2x $, so $ v'(x) = -2 $ (the derivative of $ 3 - 2x $).

Now substitute these values into the quotient rule formula:

\[
f'(x) = \frac{(3 - 2x)(0) - (1)(-2)}{(3 - 2x)^2}
\]

Simplify:

\[
f'(x) = \frac{0 + 2}{(3 - 2x)^2} = \frac{2}{(3 - 2x)^2}
\]

### Step 3: Evaluate the derivative at $x = -1$
Now, substitute $x = -1$ into the derivative:

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

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

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

Would you like further explanations?

Here are 5 related questions:
1. What is the quotient rule and how is it derived?
2. How do you find the derivative of more complex rational functions?
3. How can you verify the result using a different method like implicit differentiation?
4. Can you compute the derivative of a function without any rules, using the limit definition?
5. How does the behavior of the function $ f(x) $ change as $ x $ approaches values where the denominator tends to zero?

**Tip:** The quotient rule is especially useful when the function is expressed as a ratio of two other functions, and it avoids needing to rewrite the function using negative exponents.

find the derivative of the function $$ f(x) = \frac{1}{3 - 2x} $$ on the point (-1, 1/5) without using the chain rule

Learn how to find the derivative of the function f(x) = 1 / (3 - 2x) at the point (-1, 1/5) without using the chain rule. This problem covers key calculus concepts such as the quotient rule, differentiation, and evaluating at a specific point. Suitable for students in advanced calculus (Grades 11-12).

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