(2) \( \frac{9 x-4 y}{3 x^{2}+2 y^{\frac{1}{2}}} \)

You’ve provided the expression: \[ \frac{9x - 4y}{3x^{2} + 2\sqrt{y}} \] To assist you effectively, I'll explore a common operation performed on such expressions: **finding the derivative with respect to \( x \)**. If you had a different operation in mind (like simplifying, evaluating, etc.), please let me know! Assuming \( y \) is a function of \( x \) (i.e., \( y = y(x) \)), we'll use the **quotient rule** for differentiation. The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), its derivative is: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^{2}} \] Let's apply this to your expression. ### Step 1: Identify \( u \) and \( v \) \[ u = 9x - 4y \] \[ v = 3x^{2} + 2\sqrt{y} \] ### Step 2: Find \( u' \) and \( v' \) Differentiate \( u \) and \( v \) with respect to \( x \): \[ u' = \frac{d}{dx}(9x - 4y) = 9 - 4\frac{dy}{dx} \] \[ v' = \frac{d}{dx}\left(3x^{2} + 2\sqrt{y}\right) = 6x + 2 \cdot \frac{1}{2}y^{-\frac{1}{2}} \cdot \frac{dy}{dx} = 6x + y^{-\frac{1}{2}} \frac{dy}{dx} \] ### Step 3: Apply the Quotient Rule \[ \frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{(9 - 4\frac{dy}{dx})(3x^{2} + 2\sqrt{y}) - (9x - 4y)(6x + y^{-\frac{1}{2}} \frac{dy}{dx})}{(3x^{2} + 2\sqrt{y})^{2}} \] ### Step 4: Simplify (If Possible) This derivative can be left in this form, but depending on the relationship between \( y \) and \( x \), further simplification might be possible. If \( y \) is explicitly defined in terms of \( x \), you can substitute that relationship to simplify the expression further. ### Example: If \( y \) is Constant If \( y \) is a constant (i.e., doesn’t change with \( x \)), then \( \frac{dy}{dx} = 0 \). Substitute this into the derivative: \[ u' = 9 \] \[ v' = 6x \] So the derivative simplifies to: \[ \frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{9(3x^{2} + 2\sqrt{y}) - (9x - 4y)(6x)}{(3x^{2} + 2\sqrt{y})^{2}} \] Simplify the numerator: \[ 27x^{2} + 18\sqrt{y} - 54x^{2} + 24xy = -27x^{2} + 24xy + 18\sqrt{y} \] So, \[ \frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{-27x^{2} + 24xy + 18\sqrt{y}}{(3x^{2} + 2\sqrt{y})^{2}} \] ### Conclusion Without additional context or specific instructions, I've demonstrated how to differentiate the given expression with respect to \( x \). If you intended a different operation or have more details about \( y \)'s relationship with \( x \), feel free to provide them!