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Unformatted text preview: Section 15.5 The Chain Rule How to differentiate compositions of functions Recall that in single variable calculus, if f ( g ( x )) is a composition of functions, then df ( g ( x )) dx = f ( g ( x )) g ( x ). For functions of more than one variable, this is no longer directly the case since there are many different variables we could differentiate with respect to. We need to generalize the idea of the chain rule to functions of more than one variable. 1. Tree Diagram In order to develop the chain rule, we need a new idea called a tree diagram. We set it up as follows: Suppose f is a function of n variables, call them x 1 , . . . , x n . Suppose in addition, the variables x 1 , . . . , x n are also func- tions of m other variables, call them u 1 , . . . , u m (and we could continue letting u 1 , . . ., u m depend upon other variables etc). Then we can construct a diagram which reflects these depen- dencies. On the top row of the diagram, we write f On the the next row we write the variables f directly depends upon ( x 1 , . . . , x n ). On the the next row the variables x 1 , . . . , x n depend upon ( u 1 , . . . , u m ). We continue to do this until all variables are written down....
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