Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$?$n^{th}$ derivative of $cot x$What is the...
Is there a term for accumulated dirt on the outside of your hands and feet?
Geography in 3D perspective
What does "mu" mean as an interjection?
Wrapping homogeneous Python objects
Print last inputted byte
Is honey really a supersaturated solution? Does heating to un-crystalize redissolve it or melt it?
Why is indicated airspeed rather than ground speed used during the takeoff roll?
Turning a hard to access nut?
What are substitutions for coconut in curry?
Do I need to be arrogant to get ahead?
How are passwords stolen from companies if they only store hashes?
Am I eligible for the Eurail Youth pass? I am 27.5 years old
Have the tides ever turned twice on any open problem?
Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?
World War I as a war of liberals against authoritarians?
What does Deadpool mean by "left the house in that shirt"?
Could Sinn Fein swing any Brexit vote in Parliament?
gerund and noun applications
What is the significance behind "40 days" that often appears in the Bible?
Suggestions on how to spend Shaabath (constructively) alone
Are dual Irish/British citizens bound by the 90/180 day rule when travelling in the EU after Brexit?
Help rendering a complicated sum/product formula
Relation between independence and correlation of uniform random variables
What does Jesus mean regarding "Raca," and "you fool?" - is he contrasting them?
Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$?
$n^{th}$ derivative of $cot x$What is the proof of integral theorem i.e area under curve is given by anti derivative?Struggling to find the second derivative of this function's first derivativelargest interval over which the general solution is definedA fence of $y$ ft is $x$ ft from a wall, find shortest ladder using trigonometryDouble derivative w.r.t x and y neededQuick question regarding $int frac{cos x + sin x}{sin 2x},dx$Find the derivative of: $(f^{-1})'(0)$ with $f(x)=int_{1}^{x}cos(cos(t))dt$derivative of integralFind the $lim_{n to 0} (cos(x)+tan^2(x))^{csc(x)}$Differentiating $x = sec^2{3y}$ for $d^2y/dx^2$
$begingroup$
Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$ ?
I am introduced to $n^{th}$ differentiation and leibniz theorem as a first year undergraduate.
But what i observe is there are some explicit formula for the $n^{th}$ derivative of $sin x$,$cos x$,$log(ax+b)$,$frac{1}{ax+b}$, etc but why there isn't any method given for $sec x$, $csc x$ etc.
If the highlighted question can be answered then it is possible to find the $n^{th}$ derivative of secx.
How ?
$$y=sec x=frac{1}{cos x}=frac{2 e^{ix}}{e^{2ix}+1}$$
Now one can use lebniz theorem of $n^{th}$ differentiation to find the n derivative of product of two functions whose general derivative is known. Now here $e^{ix}$ whose derivative can be found easily. The latter part $frac{1}{e^{2ix+1}}$ is not known.
calculus algebra-precalculus derivatives
asked Mar 11 at 19:17
M DesmondM Desmond
1637
$endgroup$
add a comment |
$begingroup$
Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$ ?
I am introduced to $n^{th}$ differentiation and leibniz theorem as a first year undergraduate.
But what i observe is there are some explicit formula for the $n^{th}$ derivative of $sin x$,$cos x$,$log(ax+b)$,$frac{1}{ax+b}$, etc but why there isn't any method given for $sec x$, $csc x$ etc.
If the highlighted question can be answered then it is possible to find the $n^{th}$ derivative of secx.
How ?
$$y=sec x=frac{1}{cos x}=frac{2 e^{ix}}{e^{2ix}+1}$$
Now one can use lebniz theorem of $n^{th}$ differentiation to find the n derivative of product of two functions whose general derivative is known. Now here $e^{ix}$ whose derivative can be found easily. The latter part $frac{1}{e^{2ix+1}}$ is not known.
calculus algebra-precalculus derivatives
asked Mar 11 at 19:17
M DesmondM Desmond
1637
$endgroup$
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21
add a comment |
$begingroup$
Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$ ?
I am introduced to $n^{th}$ differentiation and leibniz theorem as a first year undergraduate.
But what i observe is there are some explicit formula for the $n^{th}$ derivative of $sin x$,$cos x$,$log(ax+b)$,$frac{1}{ax+b}$, etc but why there isn't any method given for $sec x$, $csc x$ etc.
If the highlighted question can be answered then it is possible to find the $n^{th}$ derivative of secx.
How ?
$$y=sec x=frac{1}{cos x}=frac{2 e^{ix}}{e^{2ix}+1}$$
Now one can use lebniz theorem of $n^{th}$ differentiation to find the n derivative of product of two functions whose general derivative is known. Now here $e^{ix}$ whose derivative can be found easily. The latter part $frac{1}{e^{2ix+1}}$ is not known.
calculus algebra-precalculus derivatives
asked Mar 11 at 19:17
M DesmondM Desmond
1637
$endgroup$
Is it possible to find $n^{th}$ derivative of $frac{1}{e^{ax}+b}$ ?
I am introduced to $n^{th}$ differentiation and leibniz theorem as a first year undergraduate.
But what i observe is there are some explicit formula for the $n^{th}$ derivative of $sin x$,$cos x$,$log(ax+b)$,$frac{1}{ax+b}$, etc but why there isn't any method given for $sec x$, $csc x$ etc.
If the highlighted question can be answered then it is possible to find the $n^{th}$ derivative of secx.
How ?
$$y=sec x=frac{1}{cos x}=frac{2 e^{ix}}{e^{2ix}+1}$$
Now one can use lebniz theorem of $n^{th}$ differentiation to find the n derivative of product of two functions whose general derivative is known. Now here $e^{ix}$ whose derivative can be found easily. The latter part $frac{1}{e^{2ix+1}}$ is not known.
calculus algebra-precalculus derivatives
calculus algebra-precalculus derivatives
asked Mar 11 at 19:17
M DesmondM Desmond
1637
asked Mar 11 at 19:17
M DesmondM Desmond
1637
edited Mar 11 at 19:26
M Desmond
asked Mar 11 at 19:17
M DesmondM Desmond
1637
asked Mar 11 at 19:17
M DesmondM Desmond
1637
asked Mar 11 at 19:17
M DesmondM Desmond
1637
1637
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21
add a comment |
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$$dfrac{d^n}{dx^n} f(x) = frac{a^n P_n(e^{ax})}{(e^{ax}+b)^{n+1}}$$
where $P_n$ is a polynomial of degree $n$, with
$P_0(t) = 1$ and
$$ P_{n+1}(t) = t (b + t) P_n'(t) - (n+1) t P(t)$$
I don't think there's a "closed form", unless you count Faà_di_Bruno's formula
edited Mar 11 at 19:54
user
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
$endgroup$
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144103%2fis-it-possible-to-find-nth-derivative-of-frac1eaxb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$dfrac{d^n}{dx^n} f(x) = frac{a^n P_n(e^{ax})}{(e^{ax}+b)^{n+1}}$$
where $P_n$ is a polynomial of degree $n$, with
$P_0(t) = 1$ and
$$ P_{n+1}(t) = t (b + t) P_n'(t) - (n+1) t P(t)$$
I don't think there's a "closed form", unless you count Faà_di_Bruno's formula
edited Mar 11 at 19:54
user
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
$endgroup$
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
add a comment |
$begingroup$
$$dfrac{d^n}{dx^n} f(x) = frac{a^n P_n(e^{ax})}{(e^{ax}+b)^{n+1}}$$
where $P_n$ is a polynomial of degree $n$, with
$P_0(t) = 1$ and
$$ P_{n+1}(t) = t (b + t) P_n'(t) - (n+1) t P(t)$$
I don't think there's a "closed form", unless you count Faà_di_Bruno's formula
edited Mar 11 at 19:54
user
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
$endgroup$
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
add a comment |
$begingroup$
$$dfrac{d^n}{dx^n} f(x) = frac{a^n P_n(e^{ax})}{(e^{ax}+b)^{n+1}}$$
where $P_n$ is a polynomial of degree $n$, with
$P_0(t) = 1$ and
$$ P_{n+1}(t) = t (b + t) P_n'(t) - (n+1) t P(t)$$
I don't think there's a "closed form", unless you count Faà_di_Bruno's formula
edited Mar 11 at 19:54
user
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
$endgroup$
$$dfrac{d^n}{dx^n} f(x) = frac{a^n P_n(e^{ax})}{(e^{ax}+b)^{n+1}}$$
where $P_n$ is a polynomial of degree $n$, with
$P_0(t) = 1$ and
$$ P_{n+1}(t) = t (b + t) P_n'(t) - (n+1) t P(t)$$
I don't think there's a "closed form", unless you count Faà_di_Bruno's formula
edited Mar 11 at 19:54
user
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
edited Mar 11 at 19:54
user
5,41411030
edited Mar 11 at 19:54
user
5,41411030
edited Mar 11 at 19:54
user
5,41411030
5,41411030
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
answered Mar 11 at 19:47
Robert IsraelRobert Israel
327k23216469
327k23216469
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
add a comment |
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
$begingroup$
(+1)Thanks sir, can you recommend any references where i can find the proof of Faa di bruno's formula. I have never heard of this before.
$endgroup$
– M Desmond
Mar 12 at 15:16
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144103%2fis-it-possible-to-find-nth-derivative-of-frac1eaxb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
If $y=frac{1}{e^{ax}+b}$, then $y'= a b y^2 - a y$. Perhaps this helps.
$endgroup$
– lhf
Mar 12 at 0:16
$begingroup$
How good is that , differentiation of $n$ times of this will yield its n-th derivative in terms of n-1th or less.
$endgroup$
– M Desmond
Mar 12 at 15:20
$begingroup$
For $cot x$, see math.stackexchange.com/questions/1701540/…
$endgroup$
– lhf
Mar 13 at 21:21