Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional derivatives
Lecture 32 :  Chain rules [Section 32.1]
 

Let

Then, by chain rule, is differentiable for all and we have

(ii)
 Let
         
and
      

For

we have by chain rule,
                ,
               
(iii)

Let

 
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