Question about calculating the gradient of a composite function r = ( x , y ) = x i + y j ‖ r ‖ = x 2 + y 2 Let's assume that r ≠ 0We now define f ( x , y ) = r m What is the right expression for ∇ f?1. m r m − 1 r 2. m r m − 2 r 3. m r m − 1 4. m r 0.5 m − 1 r Reasoning was that it should be number 1:the derivative of f according to x comes out as m r m − 1 ( x x 2 + y 2 ), while the derivative of f according to y comes out as m r m − 1 ( y x 2 + y 2 ). x 2 + y 2 = ‖ r ‖ = 1, and therefore this is equal to m r m − 1 r .

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Question about calculating the gradient of a composite function      r    =  (  x  ,  y  )  =  x      i    +  y      j        ‖    r    ‖    =            x      2        +          y      2      Let&#039;s assume that       r    ≠  0We now define   f  (  x  ,  y  )  =      r    m  What is the right expression for   ∇      f?1.   m      r          m      −      1            r  2.   m      r          m      −      2            r  3.   m      r          m      −      1      4.   m      r          0.5      m      −      1            r  Reasoning was that it should be number   1:the derivative of   f according to   x comes out as   m      r          m      −      1        (      x                  x        2            +              y        2              ), while the derivative of   f according to y comes out as   m      r          m      −      1        (      y                  x        2            +              y        2              ).             x      2        +          y      2        =      ‖    r    ‖    =  1, and therefore this is equal to   m      r          m      −      1            r  .

Phoebe Medina · Accepted Answer

Nothing in what you have written suggests that   ‖  r  ‖  =  1. What you have shown is that            ∂      f              ∂      x        =  m      r          m      −      1            x    r    =  m      r          m      −      2        xand            ∂      f              ∂      y        =  m      r          m      −      1            y    r    =  m      r          m      −      2        y  .So,  ∇  f  =            ∂      f              ∂      x            i    +            ∂      f              ∂      y            j    =  ⋯

Question about calculating the gradient of a composite function r=(x,y)=xi+yj ∥r∥=sqrt(x^2+y^2) Let's assume that r≠0 We now define f(x,y)=rm What is the right expression for ∇f?

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