Differentiable Functions with equality
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I was wondering if someone would be able to help me understand how to approach this question, as I'm unsure what it's saying.
The question is:
You are given a differentiable function $f$ : $R to R$ which satisfies the equality, $$f(exp(f(x))=f(x)$$
for all $x in R$. Show that if $f'(x)$ doesn't equal 0, then
$$f'(exp(f(x)))=exp(-f(x))$$
Note: This is meant to be $e$ to the power of $f(x)$ on the LHS, but I can't get it to write that, and $e$ to the power of $-f(x)$ on the RHS.
Any help with this questions would be most appreciated. Thank you.
functions derivatives
edited Nov 17 at 18:34
rafa11111
952417
asked Nov 17 at 18:14
The Statistician
92111
add a comment |
up vote
0
down vote
favorite
I was wondering if someone would be able to help me understand how to approach this question, as I'm unsure what it's saying.
The question is:
You are given a differentiable function $f$ : $R to R$ which satisfies the equality, $$f(exp(f(x))=f(x)$$
for all $x in R$. Show that if $f'(x)$ doesn't equal 0, then
$$f'(exp(f(x)))=exp(-f(x))$$
Note: This is meant to be $e$ to the power of $f(x)$ on the LHS, but I can't get it to write that, and $e$ to the power of $-f(x)$ on the RHS.
Any help with this questions would be most appreciated. Thank you.
functions derivatives
edited Nov 17 at 18:34
rafa11111
952417
asked Nov 17 at 18:14
The Statistician
92111
Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I was wondering if someone would be able to help me understand how to approach this question, as I'm unsure what it's saying.
The question is:
You are given a differentiable function $f$ : $R to R$ which satisfies the equality, $$f(exp(f(x))=f(x)$$
for all $x in R$. Show that if $f'(x)$ doesn't equal 0, then
$$f'(exp(f(x)))=exp(-f(x))$$
Note: This is meant to be $e$ to the power of $f(x)$ on the LHS, but I can't get it to write that, and $e$ to the power of $-f(x)$ on the RHS.
Any help with this questions would be most appreciated. Thank you.
functions derivatives
edited Nov 17 at 18:34
rafa11111
952417
asked Nov 17 at 18:14
The Statistician
92111
I was wondering if someone would be able to help me understand how to approach this question, as I'm unsure what it's saying.
The question is:
You are given a differentiable function $f$ : $R to R$ which satisfies the equality, $$f(exp(f(x))=f(x)$$
for all $x in R$. Show that if $f'(x)$ doesn't equal 0, then
$$f'(exp(f(x)))=exp(-f(x))$$
Note: This is meant to be $e$ to the power of $f(x)$ on the LHS, but I can't get it to write that, and $e$ to the power of $-f(x)$ on the RHS.
Any help with this questions would be most appreciated. Thank you.
functions derivatives
functions derivatives
edited Nov 17 at 18:34
rafa11111
952417
asked Nov 17 at 18:14
The Statistician
92111
edited Nov 17 at 18:34
rafa11111
952417
asked Nov 17 at 18:14
The Statistician
92111
edited Nov 17 at 18:34
rafa11111
952417
edited Nov 17 at 18:34
rafa11111
952417
edited Nov 17 at 18:34
rafa11111
952417
952417
asked Nov 17 at 18:14
The Statistician
92111
asked Nov 17 at 18:14
The Statistician
92111
asked Nov 17 at 18:14
The Statistician
92111
92111
Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36
add a comment |
Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36
Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36
add a comment |
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Apply the chain rule. Try writing $e^x$ as $exp(x)$.
– lzralbu
Nov 17 at 18:27
So am I meant to differentiate using the chain rule f(e^f(x)). So let u = f(a), and let v = a, where a = e^f(x)?
– The Statistician
Nov 17 at 18:31
Yes, differentiate the function $f circ exp circ f$.
– lzralbu
Nov 17 at 18:36