We want to differentiate the function:

$y = \frac{x+2}{(x-3)^3}$

This function can be written as a quotient of two functions, so we will apply the quotient rule, and within that, we will also use the chain rule for the differentiation of $(x-3)^3$.

Quotient Rule:

The quotient rule for differentiation states:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

In our case:

• $f(x) = x + 2$
• $g(x) = (x - 3)^3$

Step 1: Differentiate $f(x) = x + 2$

$f'(x) = 1$

Step 2: Differentiate $g(x) = (x - 3)^3$ using the chain rule

To differentiate this, we first recognize that we have a composite function. We need to differentiate the outer function and then multiply by the derivative of the inner function.

The derivative of $(x-3)^3$ is:

$g'(x) = 3(x-3)^2 \cdot \frac{d}{dx}(x - 3) = 3(x - 3)^2 \cdot 1 = 3(x - 3)^2$

Step 3: Apply the Quotient Rule

Now we can apply the quotient rule:

$\frac{d}{dx} \left( \frac{x+2}{(x-3)^3} \right) = \frac{(1) \cdot (x - 3)^3 - (x + 2) \cdot 3(x - 3)^2}{(x - 3)^6}$

Step 4: Simplify the Expression

Factor out $(x - 3)^2$ from both terms in the numerator:

$= \frac{(x - 3)^2 \left[ (x - 3) - 3(x + 2) \right]}{(x - 3)^6}$

Now, simplify the expression inside the brackets:

$(x - 3) - 3(x + 2) = x - 3 - 3x - 6 = -2x - 9$

Thus, the derivative becomes:

$\frac{(x - 3)^2 (-2x - 9)}{(x - 3)^6}$

Finally, cancel out $(x - 3)^2$ from the numerator and denominator:

$= \frac{-2x - 9}{(x - 3)^4}$

Final Answer:

$\frac{d}{dx} \left( \frac{x+2}{(x-3)^3} \right) = \frac{-2x - 9}{(x - 3)^4}$

Would you like further details or clarifications on any step?

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Here are 5 related questions to explore:

1. How do you apply the chain rule in the context of higher-degree polynomials?
2. What are the key differences between the product rule and the quotient rule in differentiation?
3. Can you explain the step-by-step approach to simplifying complex fractions after applying the quotient rule?
4. How would you handle higher-order derivatives for the function $y = \frac{x + 2}{(x - 3)^3}$?
5. How do you find the second derivative of rational functions like this one?

Tip: Always double-check your factoring steps when simplifying expressions involving exponents; it's easy to miss cancellation that can simplify the final answer.