Use the Chain Rule to find the derivative of the function \[ y=\left(x^{3}+9 x^{2}\right)^{-1} \]

To find the derivative of the function \( y = \left(x^{3} + 9x^{2}\right)^{-1} \) using the Chain Rule, follow these steps: 1. **Identify the Outer Function and the Inner Function:** - **Outer Function:** \( f(u) = u^{-1} \) - **Inner Function:** \( u = x^{3} + 9x^{2} \) 2. **Differentiate the Outer Function with Respect to the Inner Function:** \[ \frac{d}{du} f(u) = \frac{d}{du} \left(u^{-1}\right) = -u^{-2} \] 3. **Differentiate the Inner Function with Respect to \( x \):** \[ \frac{du}{dx} = \frac{d}{dx} \left(x^{3} + 9x^{2}\right) = 3x^{2} + 18x \] 4. **Apply the Chain Rule:** \[ \frac{dy}{dx} = \frac{d}{du} f(u) \cdot \frac{du}{dx} = -u^{-2} \cdot (3x^{2} + 18x) \] 5. **Substitute back the Inner Function \( u \):** \[ \frac{dy}{dx} = -\frac{3x^{2} + 18x}{\left(x^{3} + 9x^{2}\right)^{2}} \] 6. **Simplify (if desired):** You can factor out common terms for simplification: \[ \frac{dy}{dx} = -\frac{3x(x + 6)}{x^{4}(x + 9)^{2}} \] However, it's often acceptable to leave the derivative in the form obtained in step 5. **Final Answer:** \[ y' = \frac{ -\left( 3x^{2} + 18x \right) }{ \left( x^{3} + 9x^{2} \right)^{2} } \]