Chain Rule Calculus

    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.

(1) Proposition (Chain Rule Calculus) Let be a differentiable function of and , and let and be differentiable functions of . Then is a differentiable function of and

(2) Example (Chain Rule Calculus) If where and , find

    Solution. The chain rule gives,

(3) Example (Chain Rule Calculus) 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

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

(5) Example (Chain Rule Calculus) If where and , find and

    Solution. Applying the chain rule, we obtain







and  

(6) Proposition (Chain Rule Calculus) If is a differentiable function of the variables which in turn are differentiable functions of parameters then the composite function is differentiable and

(7) Example (Chain Rule Calculus) Write out the chain rule for the case for the case when and   where and

    Solution. The chain rule for the case when and yields the following the partial derivatives:


  
and  

(8) Example (Chain Rule Calculus) If where and find the value of when and

    Solution. By the chain rule,





  When and we have and so
  

(9) Example (Chain Rule Calculus)  If is differentiable and , show that

    Solution. Let , so Then according to the chain rule,



and



so that


  

(10) Example (Chain Rule Calculus)  Find if , where and   

    Solution. We have,  

(11) Example (Chain Rule Calculus) If is differentiable where and then find   

  

    Solution. We compute,







Similarly,  







and


  

  

so  

(12) Example (Chain Rule Calculus) The Cauchy-Riemann equations are



where and Show that if and are expressed in terms of polar coordinates, the Cauchy-Riemann equations become

  

    Solution. Using and we can state the chain rule to be used:      



By the chain rule,  



and  



Substituting,  



we obtain      



and so

  

Similarly, the chain rule to be used:  



By the chain rule,



and  



Substituting,



we obtain  



and also  

(13) Example (Chain Rule Calculus) If where and show that  



    Solution. By the chain rule, we have



and



Therefore,  



and



Also,  









Finally,  

(14) Example (Chain Rule Calculus) If where show that  



    Solution. By the chain rule, we have



and



Therefore,



and  

   

Also,  









Finally,

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Chain Rule Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/chain-rule-calculus.html