Use the Chain Rule to find the indicated partial...

Use the Chain Rule to find the indicated partial derivatives.
$$
\begin{array}{l}
z=x^{4}+x^{2} y, \quad x=s+2 t-u, \quad y=s t u^{2} \\
\frac{\partial z}{\partial s}, \frac{\partial z}{\partial t}, \frac{\partial z}{\partial u} \quad \text { when } s=4, t=2, u=1
\end{array}
$$

Key Concepts

Chain Rule

The Chain Rule in multivariable calculus is used to differentiate composite functions—that is, functions in which one variable depends on others that in turn depend on additional variables. It allows one to systematically compute the derivative of a dependent variable with respect to an independent variable by summing over the contributions of all intermediate variables.

Partial Differentiation

Partial differentiation involves computing the derivative of a function with respect to one variable while holding other variables constant. This concept is crucial when dealing with functions that depend on multiple variables, and it underpins the calculation of derivatives in situations where functions are composed of other functions.

Function Composition in Differentiation

Function composition in the context of differentiation refers to the scenario where a function is applied to the results of other functions. In chain rule applications, understanding how the inner functions contribute to the derivative of the outer function is essential, as each inner function’s derivative is multiplied by the derivative of the outer function with respect to that inner function.

Evaluation at a Specific Point

After applying the Chain Rule and computing the general form of the derivative, it is often necessary to evaluate the resulting expression at specific values of the independent variables. This step translates the general derivative into a concrete numerical value relevant to the problem at hand.

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