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Calculating partial derivatives involves finding the rate at which a function changes as one of its variables is altered while keeping other variables constant. When dealing with a multivariable function like \(f(x, y) = x^2 y \cos y\), we compute separate derivatives with respect to each variable.
* The partial derivative with respect to \(x\) is determined by treating \(y\) as a constant. In this case, the partial derivative \(\frac{\partial}{\partial x}(x^2 y \cos y) = 2xy \cos y\).
* Similarly, for \(y\), treat \(x\) as constant, resulting in \(\frac{\partial}{\partial y}(x^2 y \cos y)\). This can be more complex due to the involvement of multiple operations.
Understanding partial derivatives gives us insight into how each variable influences the overall function's behavior independently.

The chain rule is a key technique for differentiating compositions of functions. It becomes essential when computing partial derivatives for expressions involving nested functions.
In the function \(x^2 y \cos y\), the \(\cos y\) term requires differentiation using the chain rule. For \(y\) differentiation, we first handle the inner function \(\cos y\), which changes to \(-\sin y\) upon differentiation. This inner function's derivative is then multiplied by the outer function concerning \(y\), achieving the correct derivative for that part.
This process highlights the importance of the chain rule when differentiating compound variable terms.

The product rule helps in differentiating products of two or more functions. When applying the product rule, one takes the derivative of the first function while keeping the second fixed and vice versa, adding the two results.
The given function \(x^2 y \cos y\) is a multiplication of the three components \(x^2\), \(y\), and \(\cos y\). Focusing on \(y\) differentiation, treating \(x^2\) as a constant, and handling \(y \cos y\) with the rule:

• Derivative of \(y\) times \(\cos y\) plus
• Derivative of \(\cos y\) times \(y\) (with the derivative being \(-\sin y\))

This results in the partial derivative \(x^2 \cos y - x^2 y \sin y\). Recognizing complex products and correctly applying the product rule can simplify such differentiation tasks.