Let f ( x ) = x 3 + x. If h is the inverse function of f, then h ′ ( 2 ) ==?The derivative of f is: 3 x 2 + 1. But, how do I find the f − 1 x?

Question

Let   f  (  x  )  =      x    3    +  x. If h is the inverse function of   f, then       h    ′    (  2  )  ==?The derivative of   f is:   3      x    2    +  1. But, how do I find the       f          −      1        x?

bluayu0y · Accepted Answer

Hint: you have   y  =  f  (  x  ) and the formula      h    ′    (  y  )  =      1                  f        ′            (      x      )        .To find       h    ′    (  2  ) all you need is the value   x that solves   2  =  f  (  x  ), i.e.   2  =      x    3    +  x.

Chaz Blair · Accepted Answer

You know that   f  (      f          −      1        (  x  )  )  =  x. Applying the chain rule, you get       f    ′    (      f          −      1        (  x  )  )  (      f          −      1        (  x  )      )    ′    =  1. The expression still equals   1 when evaluated at   x  =  2. Knowing that   f  =  2 when   x  =  1, if   h is the inverse of   f, then   h  (  2  )  =  1. Now you have       f    ′    (  1  )      h    ′    (  2  )  =  1, so you can solve for       h    ′    (  2  ).

Let f ( x ) = x 3 </msup> + x . If h is the inverse function

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