What is the derivative of y = tan ( arcsin ( x ) ) ?

Question

What is the derivative of       y    =          tan              (                  arcsin                      (            x            )                          )            ?

Lauren Cardenas · Accepted Answer

Alternatively, we know that            tan              (        θ        )              =                  sin                  (          θ          )                            cos                  (          θ          )                    And that             cos              (        θ        )              =                  1        -                              sin            2                                (            θ            )                              Thus, we can say that             tan              (        θ        )              =                  sin                  (          θ          )                                      1          -                                    sin              2                                      (              θ              )                                          For       θ    =          arcsin              (        x        )             we have            tan              (                  arcsin                      (            x            )                          )              =          x                        1          -                      x            2                              Which is simply a rational function that avoids messy trig derivates and is simple to derive with the right tricks. We have only the chain rule.      y    =          x                        1          -                      x            2                                                  d        y                    d        x              =          1                        1          -                      x            2                                      d              d        x              x    +    x          d              d        x                    (              1                              1            -                          x              2                                          )                          d        y                    d        x              =          1                        1          -                      x            2                                +    x          d              d        x                    (              1                              1            -                          x              2                                          )      Then       1    -          x      2        =    u                      d        y                    d        x              =          1                        1          -                      x            2                                +    x          d              d        u                    (              1                              u                              )                      d        u                    d        x                                d        y                    d        x              =          1                        1          -                      x            2                                +    x          (      -              1                  2                      u                                          3                2                                                        )              (      -      2      x      )                          d        y                    d        x              =          1                        1          -                      x            2                                +                            x          2                                      (          1          -                      x            2                    )                                      1            -                          x              2                                          From there it&#039;s just algebra (also, as a sidenote, that             (      1      -              x        2            )       would be in absolute value bars but since it&#039;s always positive for the range of x we can take, we don&#039;t bother with it.                    d        y                    d        x              =                            (          1          -                      x            2                    )                +                  x          2                                      (          1          -                      x            2                    )                                      1            -                          x              2                                            =          1                        (          1          -                      x            2                    )                                      1            -                          x              2                                          Or, if you prefer                    d        y                    d        x              =                  (        1        -                  x          2                )                    -                  3          2

What is the derivative of y=tan(arcsin(x))?

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