Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values ofh to find the instantaneous rate of change for the given function at the given value. f ( x ) = 7 x ln ⁡ x , x = 3

Question

Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values ofh to find the instantaneous rate of change for the given function at the given value.  f  (  x  )  =  7      x          ln      ⁡      x        ,     x  =  3

dyrni0gm · Accepted Answer

Given,   f  (  x  )  =  7      x          log      ⁡      x        ;  x  =  3The rote of change of f(x) at x=3 is given by      lim          h      →      0                  f      (      3      +      h      )      −      f      (      3      )        h    =      lim          h      →      0                  7      (      3      +      h              )                  log          ⁡          (          3          +          h          )                    −      7      ×              3                  log          ⁡          3                      h  which is 0/0 formNow we apply the L-Hopital ruleSince       d          d      x        (      x          log      ⁡      x        )  =  [  2      x          log      ⁡      x      −      1        ]  log  ⁡  xThen let   y  =      x          log      ⁡      x      taking both side log  log  ⁡  y  =  log  ⁡  (      x          log      ⁡      x        )    =  log  ⁡  x  ⋅  log  ⁡  x    log  ⁡  y  =      log    2    ⁡  xon differentiate with respect to x      1    y              d      y              d      x        =  2  log  ⁡  x  ⋅      1    x          y    ′    =            2      y      log      ⁡      x        x          y    ′    =  2      x          log      ⁡      x        ⋅      x          −      1        log  ⁡  x        y    ′    =  2  log  ⁡  x  ⋅  [      x          log      ⁡      x      −      1        ]Now,      lim          h      →      0                  7      ×      [      2      log      ⁡      (      3      +      h      )      (      (      3      +      h              )                  log          ⁡          (          3          +          h          )          −          1          )                    ]      −      0        1      =  7  ×  2  (  log  ⁡  (  3  +  0  )  [  (  3  +  0      )          log      ⁡      (      3      +      0      )      −      1        ]    =  14  ×  (  log  ⁡  3  )  (      3          log      ⁡      3      −      1        )    =  14  ×  0.47712  ×  0.56301    =  3.7607

Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values ofh to find the instantaneous rate of change for the given function at the given value. f(x)=7x^(ln x), x=3

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