Simplify 1 x 2 + 1 − x 2 ( x 2 + 1 ) 3 / 2 I want to know why 1 x 2 + 1 − x 2 ( x 2 + 1 ) 3 / 2 can be simplified into 1 ( x 2 + 1 ) 3 / 2 I tried to simplify by rewriting radicals and fractions. I was hoping to see a clever trick (e.g. adding a clever zero, multiplying by a clever one? Quadratic completion?) 1 x 2 + 1 − x 2 ( x 2 + 1 ) 3 / 2 = = ( x 2 + 1 ) − 1 / 2 − x 2 ∗ ( x 2 + 1 ) − 3 / 2 = ( x 2 + 1 ) − 1 / 2 ∗ ( 1 − x 2 ∗ ( x 2 + 1 ) − 1 ) = . . . To give a bit more context, I was calculating the derivative of x x 2 + 1 in order to use newtons method for approximating the roots.

Question

Simplify       1                  x        2            +      1        −            x      2              (              x        2            +      1              )                  3                      /                    2                    I want to know why      1                  x        2            +      1        −            x      2              (              x        2            +      1              )                  3                      /                    2                    can be simplified into      1          (              x        2            +      1              )                  3                      /                    2                    I tried to simplify by rewriting radicals and fractions. I was hoping to see a clever trick (e.g. adding a clever zero, multiplying by a clever one? Quadratic completion?)                              1                                    x              2                        +            1                          −                              x            2                                (                          x              2                        +            1                          )                              3                                  /                                2                                                                        =                                          =        (                  x          2                +        1                  )                      −            1                          /                        2                          −                  x          2                ∗        (                  x          2                +        1                  )                      −            3                          /                        2                                                            =        (                  x          2                +        1                  )                      −            1                          /                        2                          ∗        (        1        −                  x          2                ∗        (                  x          2                +        1                  )                      −            1                          )                                          =        .        .        .            To give a bit more context, I was calculating the derivative of       x                  x        2            +      1       in order to use newtons method for approximating the roots.

Manuel Leach · Accepted Answer

x   is just a shorthand for       x          1              /            2      . Hence we can multiply the two halves of the first fraction in the first term by       x    2    +  1:      1                  x        2            +      1        −            x      2              (              x        2            +      1              )                  3                      /                    2                      =                    x        2            +      1              (              x        2            +      1              )                  3                      /                    2                      −            x      2              (              x        2            +      1              )                  3                      /                    2                    and the target expression follows.

Gaintentavyw4 · Accepted Answer

HINT: we have      1                  x        2            +      1        −            x      2                                x          2                +        1            3        =                    x        2            +      1      −              x        2                                      x          2                +        1            3

Simplify 1/(sqrt(x^2+1))-(x^2)/((x^2+1)^(3/2))

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