Differentiatea) 5 x 2 x 3 + 1 b) 1 1 + e x c) e x x 2

Question

Differentiatea)             5              x        2                            x        3            +      1      b)       1          1      +              e        x            c)             e      x              x      2

Addison Mueller · Accepted Answer

a)             5              x        2                            n        3            +      1      Now,       d          d      x        (            d      (      x      )              g      (      x      )        )  =            g      (      x      )              f        ′            (      x      )      −      f      (      x      )              g        ′            (      x      )              (      g      (      x      )              )        2                       (Quotient Rule)  ∴      d          d      x        [            5              x        2                            x        3            +      1        ]  =            (              x        3            +      1      )              d                  d          x                    (      5              x        2            )      −      5              x        2                    d                  d          x                    (              x        3            +      1      )              (              x        3            +      1              )        2                =            (              x        3            +      1      )      ×      5      ×      (      2      x      )      −      5              x        2                    d                  d          x                    (              x        3            +      1      )              (              x        3            +      1              )        2                =            (              x        3            +      1      )      (      10      x      )      −      15              x        4                    (              n        3            +      1              )        2                =            10              x        4            +      10      x      −      15              x        4                            x        6            +      1      +      2              n        3                =            10      x      −      5              x        4                    (              x        6            +      2              x        3            +      1      )          =            5      (      −              x        4            +      2      x      )                      x        6            +      2              x        3            +      1      b)       1          1      +              e        x            Using quotient rule,      d          d      x        (      1          1      +              e        x              )  =            (      1      +              e        x            )              d                  d          x                    (      1      )      −      1              d                  d          x                    (      1      +              e        x            )              (      1      +              e        x                    )        2                =            (      1      +              e        x            )      ×      0      −      (      0      +              e        x            )              (      1      +              e        x                    )        2                =            0      −              e        x                    (      1      +              e        x            )      62          =            −              e        x                    (      1      +              e        x                    )        2            c)             e      x              x      2            d          d      x        (            e      x              x      2        )  =                    x        2                    d                  d          x                    (              e        x            )      −              e        x                    d                  d          x                    (              x        2            )              (              x        2                    )        2                =                    x        2                    e        x            −              e        x            ×      2      x              x      4        =                    e        x            (              x        2            −      2      x      )              x      4          =                    e        x            ⋅      x      (      x      −      2      )              x      4          =                    e        x            (      x      −      2      )              x      3

Differentiate a) (5x^2)/(x^3+1) b)1/(1+e^x) c)e^x/x^2

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