If f ( x , y ) = x x x x y + ln ⁡ ( x ) [ tan − 1 ⁡ ( tan − 1 ⁡ ( tan − 1 ⁡ ( sin ⁡ ( cos ⁡ x y ) − ln ⁡ ( x + y ) ) ) ]

Question

If  f  (  x  ,  y  )  =      x                  x                              x                                          x                y                                                          +  ln  ⁡  (  x  )  [      tan          −      1        ⁡  (      tan          −      1        ⁡  (      tan          −      1        ⁡  (  sin  ⁡  (  cos  ⁡  x  y  )  −  ln  ⁡  (  x  +  y  )  )  )  ]

tatovihd1 · Accepted Answer

Let&#039;s look at the first summand, as       x          g      (      x      ,      y      )       has the derivative      x          g      (      x      ,      y      )            (          g      x        (    x    ,    y    )    log    ⁡    x    +                  g        (        x        ,        y        )            x        )  Now note that   log  ⁡  1  =  0, hence the derivative of the first summand at   (  1  ,  y  ) is      1          g      (      1      ,      y      )        ⋅            g      (      1      ,      y      )        1    =  1where we used that our   g  (  1  ,  y  )  =      1                  1                              1            y                                =  1Same trick for the second summand. It&#039;s a product, the first factor is   log  ⁡  x, now the derivative of   log  ⁡  x  ⋅  h  (  x  ,  y  ) is  log  ⁡  x  ⋅      h    x    (  x  ,  y  )  +            h      (      x      ,      y      )        x  plugging in   x  =  1 we have            h      (      1      ,      y      )        1  So the partial derivative with respect to x at   (  1  ,  y  ) of the given function is      f    x    (  1  ,  y  )  =  1  +  arctan  ⁡  arctan  ⁡  arctan  ⁡            (        sin  ⁡  (  cos  ⁡  y  )  −  log  ⁡  (  1  +  y  )            )

If f ( x , y ) = x <mrow class="MJX-TeXAtom-ORD"> x

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