Find ∂w / ∂s and ∂w / ∂t (a) by using the appropriate Chain...

Find $\partial w / \partial s$ and $\partial w / \partial t$ (a) by using the appropriate Chain Rule and (b) by converting $w$ to a function of $s$ and $t$ before differentiating.
$$w=x y z, \quad x=s+t, \quad y=s-t, \quad z=s t^{2}$$

Key Concepts

Chain Rule in Multivariable Calculus

The multivariable chain rule is a method for computing the derivative of a function that depends on intermediate variables which themselves depend on other independent variables. This technique systematically combines the rates of change along each path from the independent to the dependent variable, and is essential when differentiating composite functions in several variables.

Partial Derivatives

A partial derivative is the derivative of a multivariable function with respect to one variable while keeping the other variables constant. This concept is crucial in understanding how a function changes in a specific direction and forms the foundation for gradient, optimization, and sensitivity analysis in multivariable calculus.

Product Rule

The product rule is used when differentiating functions that are products of two or more functions. It states that the derivative of a product is the sum of each function's derivative multiplied by the product of the remaining functions. This rule is key in computing derivatives accurately when the function in question is expressed as a product of several variable-dependent terms.

Change of Variables in Differentiation

Changing variables in differentiation involves rewriting a function in terms of a new set of variables, often to simplify computation or better reflect the problem's structure. This method requires substituting the expressions of the original variables with those of the new ones before differentiating, and it is widely used in solving problems involving coordinate transformations or simplifying the form of differential equations.

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