Partial Derivatives of a Function of Two Variables

Compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the function $f(x, y) = x^2 \cdot y + \sin(y)$.

When computing the partial derivatives of a function of multiple variables, we need to consider how the function changes with respect to each variable independently. In this problem, the function given is of the form $f(x, y) = x^2 \cdot y + \sin(y)$. To find the partial derivatives, we take the derivative of the function with respect to one variable while treating the other variable as a constant.

For the partial derivative with respect to x, we treat y as a constant and differentiate the expression $x^2 \cdot y$. This simplifies to $2x \cdot y$, as the sine of y, being a function of y, acts like a constant in this differentiation. For the partial derivative with respect to y, we differentiate $x^2 \cdot y$ with respect to y, treating $x^2$ as a constant, and we also need to differentiate sine of y with respect to y. This results in $x^2 + \cos(y)$.

This problem highlights the importance of understanding partial derivatives in multivariable calculus, serving as a foundation for concepts such as gradient, divergence, and curl in vector calculus, as well as optimization problems involving several variables. Mastery of partial derivatives allows for the exploration of how functions behave in multidimensional contexts, providing insights that are crucial in fields ranging from engineering to economics.

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