Chain Rule with One Independent Parameter

    Recall that the chain rule for functions of a single variable gives the rule for differentiating a composite function: if and where and are differentiable functions, then is a a differentiable function of and



There are several versions of the chain rule for functions of more than one variable, each of them giving a rule for differentiating a composite function.

Proposition (Chain Rule with One Independent Parameter) Let be a differentiable function of and , and let and be differentiable functions of . Then is a differentiable function of and



    Proof. Because is differentiable, we can write the increment in the following form:  



where and    as both and   Dividing by we obtain  



Because and are function of , we can write their increments as  



We know that and vary continuously with , because and are differentiable, and it follows that and   as so that and   as   Therefore, we have



  
  
   
   

Example (Chain Rule with One Independent Parameter) If where and , find

    Solution. The chain rule gives,

Example (Chain Rule with Related Rates) Two objects are traveling in elliptical paths given by the following parametric equations

             and         

             and         

At what rate is the distance between the two objects changing when

    Solution. The distance between the two objects is given by



and that when we have and So



When the partial derivatives of are as follows.







    

When the derivatives of and are









So using the chain rule, we find that the distance is changing at a rate of







when

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