Use implicit differentiation to find $\frac{d y}{d x}$.
\[
\begin{array}{l}
y^{2}+3 x^{3}=9 y-7 x^{2} \\
\frac{d y}{d x}=
\end{array}
\]
So, the derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{9x^{2} + 14x}{2y - 9}}\).
Step 1 :First, we differentiate both sides of the equation with respect to \(x\).
Step 2 :Differentiating \(y^{2}\) with respect to \(x\) gives \(2y \frac{dy}{dx}\).
Step 3 :Differentiating \(3x^{3}\) with respect to \(x\) gives \(9x^{2}\).
Step 4 :Differentiating \(9y\) with respect to \(x\) gives \(9 \frac{dy}{dx}\).
Step 5 :Differentiating \(-7x^{2}\) with respect to \(x\) gives \(-14x\).
Step 6 :Putting it all together, we get \(2y \frac{dy}{dx} + 9x^{2} = 9 \frac{dy}{dx} - 14x\).
Step 7 :We can rearrange this equation to isolate \(\frac{dy}{dx}\) on one side. This gives \(\frac{dy}{dx} = \frac{9x^{2} + 14x}{2y - 9}\).
Step 8 :So, the derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{9x^{2} + 14x}{2y - 9}}\).