Find $\frac{d y}{d x}$ using implicit differentiation.
\[
3 x^{3}=5 y^{2}+9 y
\]
Answer
Final Answer: The derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{9x^2}{10y + 9}}\).
Step 1 ：Differentiate both sides of the equation \(3x^3 = 5y^2 + 9y\) with respect to \(x\).
Step 2 ：The derivative of \(3x^3\) with respect to \(x\) is \(9x^2\).
Step 3 ：The derivative of \(5y^2\) with respect to \(x\) is \(10y\frac{dy}{dx}\).
Step 4 ：The derivative of \(9y\) with respect to \(x\) is \(9\frac{dy}{dx}\).
Step 5 ：After differentiating, we get the equation \(9x^2 = 10y\frac{dy}{dx} + 9\frac{dy}{dx}\).
Step 6 ：Solve for \(\frac{dy}{dx}\) to get \(\frac{dy}{dx} = \frac{9x^2}{10y + 9}\).
Step 7 ：Final Answer: The derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{9x^2}{10y + 9}}\).
