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.
$$w=z e^{x y}, \quad x=s-t, \quad y=s+t, \quad z=s t$$

Key Concepts

Multivariable Chain Rule

This concept involves differentiating a function that depends on intermediate variables, which themselves are functions of other variables. The chain rule for multivariable functions provides a systematic way to compute the derivative by taking partial derivatives with respect to each intermediate variable and then multiplying by the derivative of that intermediate variable with respect to the target variable.

Direct Variable Substitution

This concept refers to the process of rewriting a composite function entirely in terms of a new set of independent variables before differentiating. By substituting the expressions for the intermediate variables directly into the function, one can simplify the differentiation process and compute partial derivatives directly with respect to the new variables.

Partial Differentiation of Composite Functions

This concept encompasses the techniques used to differentiate functions that depend on several variables, where those variables are themselves functions of other independent variables. It requires an understanding of how changes in the underlying independent variables propagate through the composite structure of the function to affect its overall rate of change.

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