1. **State the problem:** We are given the function $z = 3xy + 4x^2$ and asked to find the partial derivative of $z$ with respect to $y$. 2. **Recall the formula:** The partial derivative of a function with respect to one variable treats all other variables as constants. Here, we differentiate $z$ with respect to $y$, treating $x$ as a constant. 3. **Apply the derivative:** - The term $3xy$ is linear in $y$, so its derivative with respect to $y$ is $3x$. - The term $4x^2$ does not depend on $y$, so its derivative with respect to $y$ is $0$. 4. **Combine results:** $$\frac{\partial z}{\partial y} = 3x + 0 = 3x$$ 5. **Interpretation:** The partial derivative $\frac{\partial z}{\partial y}$ measures how $z$ changes as $y$ changes, keeping $x$ fixed. **Final answer:** $\boxed{3x}$