Use appropriate forms of the chain rule to find the...

Use appropriate forms of the chain rule to find the derivatives.
$$
\begin{array}{l}{\text { Let } T=x^{2} y-x y^{3}+2 ; x=r \cos \theta, y=r \sin \theta . \text { Find }} \\ {\partial T / \partial r \text { and } \partial T / \partial \theta}\end{array}
$$

Key Concepts

Multivariable Chain Rule

The multivariable chain rule is the fundamental tool used to differentiate composite functions when the variables are themselves functions of other variables. It allows us to express the derivatives of a function with respect to one variable (such as r or ?) by summing the products of the derivatives of the outer function with respect to its independent variables (x and y) and the derivatives of those independent variables with respect to the target variable. This concept is crucial when working with functions defined in one coordinate system that have been transformed into another.

Partial Differentiation

Partial differentiation involves taking the derivative of a function with several variables with respect to one variable at a time, while keeping the other variables constant. This concept is essential when dealing with functions that depend on multiple independent variables, particularly in problems requiring the computation of rates of change along specific directions or coordinate axes.

Change of Variables/Coordinate Transformation

Changing variables or coordinate transformation is a process used to rewrite a function in a different coordinate system, such as converting Cartesian coordinates to polar coordinates. This technique is employed to simplify differentiation or integration by expressing the function's variables in terms that are more natural or convenient for the problem at hand. Understanding this concept allows one to apply the chain rule effectively to compute the needed derivatives.

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