How to find the derivative of y = x ( arcsin ) ( x 2 ) ?

Question

How to find the derivative of       y    =    x          (      arcsin      )              (              x        2            )      ?

inkljvn · Accepted Answer

How do you find the derivative of   y  =  x  sin  ⁡      x    2  ?  y  =  x  a  sin  ⁡      x    2    =                    u        ⏟              u    ⋅                              a          sin          ⁡                      x            2                          ⏟              v    ⇒  y  =  u  ⋅  v  ⇒      y    ′    =      u    ′    v  +  u      v    ′    v  =  a  sin  ⁡      x    2    ⇒                              a          sin          ⁡                                                                      x                  2                                ⏞                                                    g              (              x              )                                      ⏟                    f      [      g      (      x      )      ]        =  f  [  g  (  x  )  ]  ⇒      v    ′    =      f    ′    [  g  (  x  )  ]  ⋅      g    ′    (  x  )  ←  chain rule  f  [  g  (  x  )  ]  =  a  sin  ⁡      x    2    ⇒      f    ′    [  g  (  x  )  ]  =      1          1      −      (              x        2                    )        2              ⇒      f    ′    [  g  (  x  )  ]  =      1          1      −              x        4              g  (  x  )  =      x    2    ⇒      g    ′    (  x  )  =  2  xSubstituting:      v    ′    =      f    ′    [  g  (  x  )  ]  ⋅      g    ′    (  x  )  =      1          1      −              x        4              ⋅  2  x  ⇒      v    ′    =            2      x              1      −              x        4            Substituting u,v,u&#039; and v&#039; in y&#039;:      y    ′    =      u    ′    v  +  u      v    ′    =  (  1  )  (  a  sin  ⁡      x    2    )  +  (  x  )  (            2      x              1      −              x        4              )  ⇒      y    ′    =  a  sin  ⁡      x    2    +            2              x        2                    1      −              x        4            Answer:   y  =  x  a  sin  ⁡      x    2    ⇒      y    ′    =  a  sin  ⁡      x    2    +            2              x        2                    1      −              x        4

How to find the derivative of y = x ( arcsin ) ( x 2...

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