How to differentiate sin 2 ( x 2 ) ?

Question

How to differentiate                     sin        2                    (                  x          2                )            ?

Jerimiah Short · Accepted Answer

By the chain rule.Letting       y    =                  sin        2                    (        u        )             and       u    =          x      2      , we need to differentiate both functions and multiply the derivatives together.  The derivative of       y    =                  sin        2            u       can be obtained as follows:      y    =          (              sin        u            )              (              sin        u            )      By the product rule:      y    ′    =          cos      u        ×          sin      u        +          cos      u        ×          sin      u            y    ′    =    2          cos      u              sin      u            y    ′    =          sin      2        u  The derivative of       u    =          x      2       can be obtained using the quotient rule:      u    =          x      2            u    ′    =                  1        ×        2        -        x        ×        0                    2        2                  u    ′    =          2      4            u    ′    =          1      2      Therefore, the derivative of       y    =                  sin        2                    (                  x          2                )             is:                    d        y                    d        x              =          sin      2        u    ×          1      2                          d        y                    d        x              =          sin      2              (              x        2            )        ×          1      2                          d        y                    d        x              =          1      2        ×    2    ×          sin              (                  x          2                )              ×          cos              (                  x          2                )                                d        y                    d        x              =          sin              (                  x          2                )              ×          cos              (                  x          2                )

wabennestvzz · Accepted Answer

We use the Trigo. Identity       :    1    -          cos      θ        =    2                  sin        2                    (                  θ          2                )            .Thus, in our case,                     sin        2                    (                  x          2                )              =          1      2              (      1      -              cos        x            )      . Thus,            d              d        x                    [                        sin          2                          (                      x            2                    )                    ]        =          d              d        x                    [              1        2                    (        1        -                  cos          x                )            ]        =          1      2              d              d        x                    [      1      -              cos        x            ]            =          1      2              [              d                  d          x                    1      -              d                  d          x                            cos        x            ]        =          1      2              [      0      -              (        -                  sin          x                )            ]        =          1      2              sin      x      .Note that, since,             sin      θ        =    2          sin              (                  θ          2                )                    cos              (                  θ          2                )                        d              d        x                    [                        sin          2                          (                      x            2                    )                    =              1        2                    sin        x            =              1        2                    (        2                  sin                      (                          x              2                        )                                    cos                      (                          x              2                        )                          )            =              sin                  (                      x            2                    )                            cos                  (                      x            2                    )                    ,

How do you differentiate sin^2(x/2)?

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