Chain Rule with Two Independent Parameters

Proposition (Chain Rule with Two Independent Parameters) Suppose is a differentiable function at and that the partial derivatives of and exist at Then the composite function is differentiable at with



    Proof. Fix , then the composite function depends on alone, and we have the chain rule of one independent variable. Applying this rule, we have



Similarly, if is fixed we are lead to

Example (Chain Rule with Two Independent Parameters) If where and , find and

    Solution. Applying the chain rule, we obtain







and  

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Chain Rule With Two Independent Parameters
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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