f-f-x-f-f-x-f-x-

Question Number 2035 by Rasheed Soomro last updated on 31/Oct/15

f (f^{'} (x)) = f^{'} (f (x))

f (x) = ?

Commented by Yozzi last updated on 31/Oct/15

T r y f (x) = e^{x} . f^{^{'}} (x) = e^{x} \Rightarrow f (f^{^{'}} (x)) = e^{e^{x}}

f^{^{'}} (f (x)) = e^{e^{x}}

f (x) = a^{x} \Rightarrow f^{^{'}} (x) = a^{x} l n a (a > 0)

∴ f (f^{^{'}} (x)) = a^{a^{x} l n a}

f^{^{'}} (f (x)) = a^{a^{x} l n a} l n a \neq f (f^{^{'}} (x))

Commented by 123456 last updated on 31/Oct/15

f ∙ f^{- 1} = f^{- 1} ∙ f = I (if f is invertible)

f [f^{'} (x)] = f^{'} [f (x)]

f^{'} (x) = f^{- 1} {f^{'} [f (x)]}

f (x) = {(f^{'})}^{- 1} {f [f^{'} (x)]}

Answered by 123456 last updated on 31/Oct/15

f (x) = a^{x}

f^{'} (x) = a^{x} \ln a (a > 0)

f [f^{'} (x)] = a^{a^{x} \ln a}

f^{'} [f (x)] = a^{a^{x}} \ln a

a^{a^{x} \ln a} = a^{a^{x}} \ln a

a^{a^{x} (\ln a - 1)} = \ln a (a^{a^{x}} \neq 0)

a^{x} (\ln a - 1) = \log_{a} \ln a (\ln a > 0 \Rightarrow a > 1)

a^{x} = \frac{\log_{a} \ln a}{\ln a - 1} (\ln a \neq 1 \Rightarrow a \neq e)

x = \log_{a} (\frac{\log_{a} \ln a}{\ln a - 1})

- - - - - - - - - -

a = e \Rightarrow \ln a = 1 \Rightarrow \log_{a} \ln a = 0

f (x) = e^{x} is a solution

Answered by prakash jain last updated on 31/Oct/15

f (x) = x \dots . (s o l u t i o n 1)

f^{'} (x) = 1

f (f^{'} (x)) = 1

f^{'} (f (x)) = 1

f (x) = \frac{x^{2}}{2} \dots (s o l u t i o n 2)

f^{'} (x) = x

f (f^{'} (x)) = f (x) = \frac{x^{2}}{2}

f^{'} (f (x)) = f^{'} (\frac{x^{2}}{2}) = \frac{x^{2}}{2}

Commented by prakash jain last updated on 31/Oct/15

f (x) = \frac{x^{n}}{k}

f^{'} (x) = \frac{{n x}^{n - 1}}{k}

f (f^{'} (x)) = \frac{{(\frac{{n x}^{n - 1}}{k})}^{n}}{k} = \frac{n^{n} x^{n (n - 1)}}{k^{n + 1}}

f^{'} (f (x)) = \frac{n {(\frac{x^{n}}{k})}^{n - 1}}{k} = \frac{{n x}^{n (n - 1)}}{k^{n}}

\frac{{n x}^{n (n - 1)}}{k^{n}} = \frac{n^{n} x^{n (n - 1)}}{k^{n + 1}}

k = n^{n - 1}

f (x) = \frac{x^{n}}{n^{n - 1}}

Commented by Rasheed Soomro last updated on 01/Nov/15

E x c e l l e n t! T h i s i s g e n e r a l s o l u t i o n c o v e r s

f (x) = x, f (x) = \frac{x^{2}}{2}, f (x) = \frac{x^{3}}{9} \dots . (a w h o l e c a e g a r y)