Draw a branch diagram and write a Chain Rule formula for...

Draw a branch diagram and write a Chain Rule formula for each derivative.

$\begin{array}{l}{\frac{\partial w}{\partial x} \text { and } \frac{\partial w}{\partial y} \text { for } w=f(r, s, t), \quad r=g(x, y), \quad s=h(x, y)} \\ {t=k(x, y)}\end{array}$

Key Concepts

Multivariable Chain Rule

The multivariable chain rule is an extension of the standard chain rule to functions with several independent variables. It allows the computation of partial derivatives of a composite function by summing the products of the partial derivative of the outer function with respect to each intermediate variable and the partial derivative of the corresponding intermediate variable with respect to the independent variable. This rule ensures that all paths of dependency are accounted for when differentiating with respect to a particular variable.

Partial Derivatives

Partial derivatives represent the rate of change of a multivariable function with respect to one variable while keeping other variables constant. They serve as the building blocks for applying the chain rule in multiple dimensions, allowing for the analysis of how variations in each independent component separately affect the composite function.

Dependency Diagram (Branch Diagram)

A dependency diagram, often in the form of a branch or tree diagram, visually maps out the relationships between variables in a composite function. It illustrates the different pathways through which an independent variable influences the output, thereby guiding the systematic application of the multivariable chain rule by clearly showing the required partial derivatives and their connections.

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