To find the derivative of the function $y = (3x - 5)^{-1}$, we can use the chain rule.

Step-by-step solution:

1. Rewrite the function: $y = (3x - 5)^{-1}$ This is equivalent to: $y = \frac{1}{3x - 5}$

2. Apply the chain rule: The derivative of $(3x - 5)^{-1}$ with respect to $x$ involves the chain rule, since $3x - 5$ is an inner function.

   • The outer function is $u^{-1}$, where $u = 3x - 5$.
   • The derivative of $u^{-1}$ with respect to $u$ is $-u^{-2}$.

3. Differentiate the inner function: The derivative of $3x - 5$ with respect to $x$ is $3$.

4. Combine the results using the chain rule: $\frac{dy}{dx} = -\frac{1}{(3x - 5)^2} \times 3$

5. Simplify the derivative: $\frac{dy}{dx} = -\frac{3}{(3x - 5)^2}$

Final Answer:

$\frac{dy}{dx} = -\frac{3}{(3x - 5)^2}$

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. What is the derivative of $y = (5x + 7)^{-1}$?
2. How would you differentiate $y = (2x^2 - 4x + 1)^{-1}$?
3. What is the second derivative of $y = (3x - 5)^{-1}$?
4. How does the chain rule apply when differentiating composite functions?
5. How would you find the derivative of $y = (3x - 5)^{-3}$?

Tip: When differentiating powers of functions, always remember to apply the chain rule if there's an inner function involved.