method of differentiation question...
let, \[e ^{f(x)} = \ln x\] if g(x) is the inverse of the function of f(x) , then g'(x) is..
\[\large e ^{f(x)} = \ln x\]If you take ln of both sides\[\large f(x) = \ln \left( \ln x \right)\]
You should be able to just find the inverse of it by swapping x and y\[\large x = \ln \left( \ln y \right)\] e to the power of both sides\[\large e^x = \ln y\]and e to the power of both sides again\[\Large e^{e^x} = y\] Now, remember that y is the inverse of f(x) (so y is g(x)) You can just differentiate, using the chain rule.
\[f(x)=\ln( \ln x)\] let y=f(x) y=ln(lnx) interchange x and y x=ln(lny) \[\ln y=e^x\] \[y=e ^{{e}^x}\] \[g(x)=f ^{-1}\left( x \right)=y=e ^{e ^{x}}\]
find g'(x)
\[\ln g(x)=\ln \left( e ^{e ^{x}} \right)=e^x \ln e=e^x\] diff. w.r.t.x \[\frac{ g \prime \left( x \right) }{ g(x) }=e^x\] \[g \prime(x)=g(x)e ^{x}=?\]
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