Calculation of the derivative of e cos ⁡ ( x ) from first principlesThe derivative of e cos ⁡ ( x ) is − sin ⁡ ( x ) e cos ⁡ ( x ) . However I would like to prove it using first principles, i.e. by using f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h

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Calculation of the derivative of       e          cos      ⁡      (      x      )       from first principlesThe derivative of       e          cos      ⁡      (      x      )       is   −  sin  ⁡  (  x  )      e          cos      ⁡      (      x      )      . However I would like to prove it using first principles, i.e. by using       f    ′    (  x  )  =      lim          h      →      0                  f      (      x      +      h      )      −      f      (      x      )        h

Gwendolyn Case · Accepted Answer

f    ′    (  x  )  =      lim          h      →      0                  f      (      x      +      h      )      −      f      (      x      )        h    =      e          cos      ⁡      x            lim          h      →      0                          e                  cos          ⁡          x          (          cos          ⁡          h          −          1          )          −          sin          ⁡          x          sin          ⁡          h                    −      1        h    ,we use hospital Rule to compute the limitWe have                              lim                      h            →            0                                                              e                              cos                ⁡                x                (                cos                ⁡                h                −                1                )                −                sin                ⁡                x                sin                ⁡                h                                      −            1                    h                                    =                  lim                      h            →            0                                    [          (          −          sin          ⁡          h          (          cos          ⁡          x          )          −          cos          ⁡          h          sin          ⁡          x          )                                 e                          cos              ⁡              x              (              cos              ⁡              h              −              1              )              −              sin              ⁡              x              sin              ⁡              h                                ]                                                  =                  lim                      h            →            0                          (        −        sin        ⁡        h        (        cos        ⁡        x        )        −        cos        ⁡        h        sin        ⁡        x        )                ⋅                          lim                      h            →            0                                    e                      cos            ⁡            x            (            cos            ⁡            h            −            1            )            −            sin            ⁡            x            sin            ⁡            h                                                            =        −        sin        ⁡        x        .

Calculation of the derivative of e^(cos(x)) from first principles

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