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In this diagram, the leftmost corner corresponds to Since has two independent variables , there are two lines coming from this corner. The upper branch corresponds to the variable and the lower branch corresponds to the variable Since each of these variables is then dependent on one variable one branch then comes from and one branch comes from Last, each of the branches on the far right has a label that represents the path traveled to reach that branch. The top branch is reached by following the branch, then the branch; therefore, it is labeled The bottom branch is similar: first the branch, then the branch. This branch is labeled To get the formula for add all the terms that appear on the rightmost side of the diagram. This gives us [link] .
In [link] , is a function of and both and are functions of the independent variables
Suppose and are differentiable functions of and and is a differentiable function of Then, is a differentiable function of and
and
We can draw a tree diagram for each of these formulas as well as follows.
To derive the formula for start from the left side of the diagram, then follow only the branches that end with and add the terms that appear at the end of those branches. For the formula for follow only the branches that end with and add the terms that appear at the end of those branches.
There is an important difference between these two chain rule theorems. In [link] , the left-hand side of the formula for the derivative is not a partial derivative, but in [link] it is. The reason is that, in [link] , is ultimately a function of alone, whereas in [link] , is a function of both
Calculate and using the following functions:
To implement the chain rule for two variables, we need six partial derivatives— and
To find we use [link] :
Next, we substitute and
To find we use [link] :
Then we substitute and
Now that we’ve see how to extend the original chain rule to functions of two variables, it is natural to ask: Can we extend the rule to more than two variables? The answer is yes, as the generalized chain rule states.
Let be a differentiable function of independent variables, and for each let be a differentiable function of independent variables. Then
for any
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