Gradient of a Function Involving Sine

Compute the gradient of the function $f(x, y) = x^2 \sin(y)$.

In this problem, you are tasked with finding the gradient of a function that combines elements of trigonometry and polynomial expressions. The gradient of a multivariable function is a vector that contains all of its partial derivatives. It points in the direction of the greatest rate of increase of the function, making it a crucial concept in fields such as optimization and physics.

For the given function, your primary goal is to compute the partial derivatives with respect to each variable. This requires a good understanding of how to differentiate functions that involve product and chain rules. Differentiation involving trigonometric functions, such as sine, necessitates familiarity with their specific derivative forms as well. Recognizing that partial differentiation treats other variables as constants can help simplify the process.

Understanding the gradient not only helps in building an intuition about the behavior of the function around a point but also serves as a foundational tool for more advanced topics such as Lagrange multipliers in constrained optimization. It's essential to grasp how the gradient operates within the context of higher dimensional calculus and how it applies to real-world problems.

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