Say f ( x ) = ln ⁡ ( 1 + 2 x + 2 x 2 ) or g ( x ) = tan ⁡ ( 2 x 4 − x ).Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of 1 1 + x or 1 1 − x and then manipulate it to get result

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Say  f  (  x  )  =  ln  ⁡  (  1  +  2  x  +  2      x    2    ) or   g  (  x  )  =  tan  ⁡  (  2      x    4    −  x  ).Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of             1              1        +        x             or             1              1        −        x             and then manipulate it to get result

trutdelamodej0 · Accepted Answer

keep substitution in mind. It may not give the whole infinite series, but you will usually get the first several terms. So,      1          1      +      t        =  1  −  t  +      t    2    −      t    3    +      t    4    −      t    5    ⋯  log  ⁡  (  1  +  t  )  =  t  −            t      2        2    +            t      3        3    −            t      4        4    ⋯Taking   t  =  2  x  +  2      x    2   correctly gives the first few terms of   log  ⁡  (  1  +  2  x  +  2      x    2    )  ,, up to       x    4    log  ⁡  (  1  +  2  x  +  2      x    2    )  =  2  x  −            4              x        3              3    +  2      x    4    ⋯

Say f(x)=ln(1+2x+2x^2) or g(x)=tan(2x^(4)−x).Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of 1/1+x or 1/1−x . Think of any shortcut for listed above functions.

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