1. The problem asks for the partial derivative of the function $$f(x, y) = 3x^2 + 4xy + y^2$$ with respect to $$y$$. 2. The formula for the partial derivative of a function $$f(x, y)$$ with respect to $$y$$ is $$\frac{\partial f}{\partial y}$$, which means we differentiate $$f$$ treating $$x$$ as a constant. 3. Differentiate each term: - The derivative of $$3x^2$$ with respect to $$y$$ is $$0$$ because $$x^2$$ is constant with respect to $$y$$. - The derivative of $$4xy$$ with respect to $$y$$ is $$4x$$ because $$x$$ is constant and derivative of $$y$$ is $$1$$. - The derivative of $$y^2$$ with respect to $$y$$ is $$2y$$. 4. Combine the results: $$\frac{\partial f}{\partial y} = 0 + 4x + 2y = 4x + 2y$$ 5. Therefore, the partial derivative of $$f(x, y)$$ with respect to $$y$$ is $$4x + 2y$$. 6. Comparing with the options given, the correct answer is option d. $$4x + 2y$$.