Find the first derivative of the following exponential and logarithmic functions a ) y = e 5 x b ) y = x e 2 x c ) y = 2 ( x − 1 )

Question

Find the first derivative of the following exponential and logarithmic functions  a  )     y  =      e          5      x          b  )     y  =  x      e          2      x          c  )     y  =      2          (      x      −      1      )

Raven Hawkins · Accepted Answer

d          d      x        (      e    x    )  =      e    x    ,      d          d      x        (      a    x    )  =      a    x    ln  ⁡  (  a  )        d          d      x        (  u  ⋅  v  )  =  u  ⋅            d      v              d      x        +  v  ⋅            d      u              d      x        a  )  y  =      e          5      x                    d      y              d      x        =      d          d      x        (      e          5      x        )  =      e          5      x        ⋅      d          d      x        (  5  x  )    =      e          5      x        ⋅  5  =  5      e          5      x        b  )  y  =  x      e          2      x                    d      y              d      x        =  x  ⋅      d          d      x        (      e          2      x        )  +      e          2      x        ⋅      d          d      x        (  x  )    =  x  ⋅  2  ⋅      e          2      x        +      e          2      x        ⋅  1    =  2  x      e          2      x        +      e          2      x        =  (  2  x  +  1  )      e          2      x

Rosemary Chase · Accepted Answer

c  )  y  =      2          (      x      −      1      )        =      1    2    ⋅      2    x                d      y              d      x        =      1    2    ⋅      d          d      x        (      2    x    )    =      1    2    ⋅      2    x    ⋅  ln  ⁡  (  2  )    =  ln  ⁡  (  2  )  ⋅      2          x      −      1

Find the first derivative of the following exponential and logarithmic functions a) y=e^(5x) b) y=xe^(2x) c) y=2^((x-1))

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