find \( \frac{d y}{d x} \) 3) \( y=\frac{\left(3 x^{2}-5 x\right)^{10}}{\sqrt{x}} \)

To find \( \frac{dy}{dx} \) for the function \( y = \frac{(3x^2 - 5x)^{10}}{\sqrt{x}} \), we can use the quotient rule along with the chain rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then: \[ \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \] In this case, let \( u = (3x^2 - 5x)^{10} \) and \( v = \sqrt{x} \). First, we need to find \( \frac{du}{dx} \) using the chain rule: \[ \frac{du}{dx} = 10(3x^2 - 5x)^9 \cdot (6x - 5) \] Next, we calculate \( \frac{dv}{dx} \): \[ \frac{dv}{dx} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}} \] Now we compile the pieces into the quotient rule: \[ \frac{dy}{dx} = \frac{\sqrt{x} \cdot (10(3x^2 - 5x)^9 (6x - 5)) - (3x^2 - 5x)^{10} \cdot \frac{1}{2\sqrt{x}}}{x} \] This computation gives us \( \frac{dy}{dx} \). Make sure to simplify the final expression to get your answer in the desired form.