Differentiation- proof by Induction The 2019 Stack Overflow Developer Survey Results Are InDifferentiation help required!Help with how to prepare the inductive step of a strong induction exercise.Find the number of flags of different types using inductionUse induction to prove that $F_n ge sqrt 2 ^n$ for $n ge 6$Fibonacci Sequence: Prove $f_1+f_3+dots+f_2n-1=f_2n$ by Induction.Inductively proving a fibonacci numbers statementFind a conjecture for $F_1+F_2+…+F_n$Strong Inductive proof for inequality using Fibonacci sequenceFinding $f(x)$ when given a composite function?is it possible to find a closed expression to this?
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Differentiation- proof by Induction
The 2019 Stack Overflow Developer Survey Results Are InDifferentiation help required!Help with how to prepare the inductive step of a strong induction exercise.Find the number of flags of different types using inductionUse induction to prove that $F_n ge sqrt 2 ^n$ for $n ge 6$Fibonacci Sequence: Prove $f_1+f_3+dots+f_2n-1=f_2n$ by Induction.Inductively proving a fibonacci numbers statementFind a conjecture for $F_1+F_2+…+F_n$Strong Inductive proof for inequality using Fibonacci sequenceFinding $f(x)$ when given a composite function?is it possible to find a closed expression to this?
$begingroup$
Here is my problem:
"Suppose f is a differentiable function whose domain is $(-infty,infty)$. We define an infinite sequence of functions $f_n(x)$ as follows:
$f_1(x)=f(x), f_2(x)=f(f_1(x))$, and so on.
That is,
$f_n(x)= f(f_n-1(x))$ for $ngeq 2$.
State an explicit formula for $fracddx[f_n(x)]$ in which the only derivative is $f'$ and then prove that your formula is correct using Mathematical Induction"
So far, I have found that
$fracddx f_2(x)= f'(f(x))cdot f'(x)$
$fracddx f_3(x)= f'(f(f(x)))cdot f'(f(x))cdot f'(x)$
I saw a pattern an my formula is $f'(x)cdot f'(f(x))cdot f'(f(f(x)))cdots f'(f(dots(f(x))dots))$ for as large n is.
I am not sure how to prove this using induction though....
Thanks
calculus induction chain-rule
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
$endgroup$
add a comment |
$begingroup$
Here is my problem:
"Suppose f is a differentiable function whose domain is $(-infty,infty)$. We define an infinite sequence of functions $f_n(x)$ as follows:
$f_1(x)=f(x), f_2(x)=f(f_1(x))$, and so on.
That is,
$f_n(x)= f(f_n-1(x))$ for $ngeq 2$.
State an explicit formula for $fracddx[f_n(x)]$ in which the only derivative is $f'$ and then prove that your formula is correct using Mathematical Induction"
So far, I have found that
$fracddx f_2(x)= f'(f(x))cdot f'(x)$
$fracddx f_3(x)= f'(f(f(x)))cdot f'(f(x))cdot f'(x)$
I saw a pattern an my formula is $f'(x)cdot f'(f(x))cdot f'(f(f(x)))cdots f'(f(dots(f(x))dots))$ for as large n is.
I am not sure how to prove this using induction though....
Thanks
calculus induction chain-rule
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
$endgroup$
1
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32
add a comment |
$begingroup$
Here is my problem:
"Suppose f is a differentiable function whose domain is $(-infty,infty)$. We define an infinite sequence of functions $f_n(x)$ as follows:
$f_1(x)=f(x), f_2(x)=f(f_1(x))$, and so on.
That is,
$f_n(x)= f(f_n-1(x))$ for $ngeq 2$.
State an explicit formula for $fracddx[f_n(x)]$ in which the only derivative is $f'$ and then prove that your formula is correct using Mathematical Induction"
So far, I have found that
$fracddx f_2(x)= f'(f(x))cdot f'(x)$
$fracddx f_3(x)= f'(f(f(x)))cdot f'(f(x))cdot f'(x)$
I saw a pattern an my formula is $f'(x)cdot f'(f(x))cdot f'(f(f(x)))cdots f'(f(dots(f(x))dots))$ for as large n is.
I am not sure how to prove this using induction though....
Thanks
calculus induction chain-rule
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
$endgroup$
Here is my problem:
"Suppose f is a differentiable function whose domain is $(-infty,infty)$. We define an infinite sequence of functions $f_n(x)$ as follows:
$f_1(x)=f(x), f_2(x)=f(f_1(x))$, and so on.
That is,
$f_n(x)= f(f_n-1(x))$ for $ngeq 2$.
State an explicit formula for $fracddx[f_n(x)]$ in which the only derivative is $f'$ and then prove that your formula is correct using Mathematical Induction"
So far, I have found that
$fracddx f_2(x)= f'(f(x))cdot f'(x)$
$fracddx f_3(x)= f'(f(f(x)))cdot f'(f(x))cdot f'(x)$
I saw a pattern an my formula is $f'(x)cdot f'(f(x))cdot f'(f(f(x)))cdots f'(f(dots(f(x))dots))$ for as large n is.
I am not sure how to prove this using induction though....
Thanks
calculus induction chain-rule
calculus induction chain-rule
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
edited Oct 28 '15 at 1:21
JMoravitz
48.8k43989
48.8k43989
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
asked Oct 28 '15 at 0:53
Priyank JainPriyank Jain
62
62
1
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32
add a comment |
1
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32
1
1
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Heh. You're un UMTYMP, right? :)
Here's a hint:
Try to find d/dx[f2(x)] first, then d/dx[f3(x)]. You should see a pattern that is provable by induction.
Edit: Write the formula using big pi notation. Assume this formula works for n=k. Now, prove that it still holds true when you plug in n=k+1 using the previous assumption.
answered Oct 28 '15 at 1:01
JedJed
719414
$endgroup$
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
add a comment |
Your Answer
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1 Answer
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1 Answer
1
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oldest
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$begingroup$
Heh. You're un UMTYMP, right? :)
Here's a hint:
Try to find d/dx[f2(x)] first, then d/dx[f3(x)]. You should see a pattern that is provable by induction.
Edit: Write the formula using big pi notation. Assume this formula works for n=k. Now, prove that it still holds true when you plug in n=k+1 using the previous assumption.
answered Oct 28 '15 at 1:01
JedJed
719414
$endgroup$
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
add a comment |
$begingroup$
Heh. You're un UMTYMP, right? :)
Here's a hint:
Try to find d/dx[f2(x)] first, then d/dx[f3(x)]. You should see a pattern that is provable by induction.
Edit: Write the formula using big pi notation. Assume this formula works for n=k. Now, prove that it still holds true when you plug in n=k+1 using the previous assumption.
answered Oct 28 '15 at 1:01
JedJed
719414
$endgroup$
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
add a comment |
$begingroup$
Heh. You're un UMTYMP, right? :)
Here's a hint:
Try to find d/dx[f2(x)] first, then d/dx[f3(x)]. You should see a pattern that is provable by induction.
Edit: Write the formula using big pi notation. Assume this formula works for n=k. Now, prove that it still holds true when you plug in n=k+1 using the previous assumption.
answered Oct 28 '15 at 1:01
JedJed
719414
$endgroup$
Heh. You're un UMTYMP, right? :)
Here's a hint:
Try to find d/dx[f2(x)] first, then d/dx[f3(x)]. You should see a pattern that is provable by induction.
Edit: Write the formula using big pi notation. Assume this formula works for n=k. Now, prove that it still holds true when you plug in n=k+1 using the previous assumption.
answered Oct 28 '15 at 1:01
JedJed
719414
edited Oct 28 '15 at 1:06
answered Oct 28 '15 at 1:01
JedJed
719414
answered Oct 28 '15 at 1:01
JedJed
719414
answered Oct 28 '15 at 1:01
JedJed
719414
719414
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
add a comment |
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
Yes I am form UMTYMP, how did you know? And I edited my question for further clarification.
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:05
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
And, I know because that's the professional problem for this week. You should ask on Moodle if you want instructors' help. I don't think I am allowed to give you the answer.
$endgroup$
– Jed
Oct 28 '15 at 1:07
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
$begingroup$
Thanks @JMoravitz that makes it much more readable
$endgroup$
– Priyank Jain
Oct 28 '15 at 1:25
add a comment |
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1
$begingroup$
Welcome to Stackexchange. You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context: What you understand about the problem, what you've tried so far, etc. Something to both show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
$endgroup$
– David
Oct 28 '15 at 0:56
$begingroup$
I recommend visiting this page in order to learn a bit about how to type mathematics on this site. I've taken the liberty of typesetting several of your equations above which helps improve the readability of the question.
$endgroup$
– JMoravitz
Oct 28 '15 at 1:23
$begingroup$
It's helpful to also define $f_0(x) = x$ — the last term in your two examples is $f'(f_0(x))$.
$endgroup$
– BrianO
Oct 28 '15 at 1:32