How do you find the derivative of an inverse function of #f(x)=x^5+3x−2#?

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Answer 1 · 2018-05-11T08:05:11

#d/(dx) (f^(-1)(x)) = 1/(5(f^(-1)(x))^4+3)#

The function #f(x)# is described by the equation:

#y = x^5+3x-2#

An equation describing its inverse can be formed by swapping #x# and #y# to get:

#x = y^5+3y-2#

Then taking the derivative with respect to #x# we find:

#1 = 5y^4 (dy)/(dx) + 3 (dy)/(dx)#

So:

#(dy)/(dx) = 1/(5y^4+3)#

That is:

#d/(dx) (f^(-1)(x)) = 1/(5(f^(-1)(x))^4+3)#