Chain rule doubt
$begingroup$
I have a doubt of appling the chain rule.
I have this $L$ function:
$$
L = ycdot log(frac{e^{a x+b}}{e^{ax+b} + exp^{cx+d}})
$$
I can rewrite it as:
$$
L = ycdot log(p)
$$
where
$$
p = frac{e^{v_{0}}}{e^{v_{0}}+e^{v_{1}}}
$$
$$
v_{0} = ax+b
$$
$$
v_{1} = cx+d
$$
If I apply the chain rule I have this:
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{partial p}{partial v_{0}} cdot frac{partial v_{0}}{partial x}
$$
But I know that I am missing somewhere the value of $frac{partial v_{1}}{partial x}$
What I am doing wrong?
derivatives chain-rule
$endgroup$
add a comment |
$begingroup$
I have a doubt of appling the chain rule.
I have this $L$ function:
$$
L = ycdot log(frac{e^{a x+b}}{e^{ax+b} + exp^{cx+d}})
$$
I can rewrite it as:
$$
L = ycdot log(p)
$$
where
$$
p = frac{e^{v_{0}}}{e^{v_{0}}+e^{v_{1}}}
$$
$$
v_{0} = ax+b
$$
$$
v_{1} = cx+d
$$
If I apply the chain rule I have this:
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{partial p}{partial v_{0}} cdot frac{partial v_{0}}{partial x}
$$
But I know that I am missing somewhere the value of $frac{partial v_{1}}{partial x}$
What I am doing wrong?
derivatives chain-rule
$endgroup$
add a comment |
$begingroup$
I have a doubt of appling the chain rule.
I have this $L$ function:
$$
L = ycdot log(frac{e^{a x+b}}{e^{ax+b} + exp^{cx+d}})
$$
I can rewrite it as:
$$
L = ycdot log(p)
$$
where
$$
p = frac{e^{v_{0}}}{e^{v_{0}}+e^{v_{1}}}
$$
$$
v_{0} = ax+b
$$
$$
v_{1} = cx+d
$$
If I apply the chain rule I have this:
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{partial p}{partial v_{0}} cdot frac{partial v_{0}}{partial x}
$$
But I know that I am missing somewhere the value of $frac{partial v_{1}}{partial x}$
What I am doing wrong?
derivatives chain-rule
$endgroup$
I have a doubt of appling the chain rule.
I have this $L$ function:
$$
L = ycdot log(frac{e^{a x+b}}{e^{ax+b} + exp^{cx+d}})
$$
I can rewrite it as:
$$
L = ycdot log(p)
$$
where
$$
p = frac{e^{v_{0}}}{e^{v_{0}}+e^{v_{1}}}
$$
$$
v_{0} = ax+b
$$
$$
v_{1} = cx+d
$$
If I apply the chain rule I have this:
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{partial p}{partial v_{0}} cdot frac{partial v_{0}}{partial x}
$$
But I know that I am missing somewhere the value of $frac{partial v_{1}}{partial x}$
What I am doing wrong?
derivatives chain-rule
derivatives chain-rule
edited Dec 13 '18 at 12:48
idea
2,15841125
2,15841125
asked Dec 13 '18 at 12:35
MaikMaik
212
212
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Note that $$p=p(v_0,v_1)\
frac{dp}{dx}=frac{partial p}{partial v_0}frac{dv_0}{dx}+frac{partial p}{partial v_1}frac{dv_1}{dx}$$
$endgroup$
add a comment |
$begingroup$
Your result is wrong because $p$ is a function of $x$ that contains the two functions $nu_0$ and $nu_1$ but not in the nested form $p(nu_0(nu_1))$. So we must write its derivative using also the quotient rule as:
$$
frac{partial p}{partial x}=frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
and the correct final result is
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
$endgroup$
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%2f3037950%2fchain-rule-doubt%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Note that $$p=p(v_0,v_1)\
frac{dp}{dx}=frac{partial p}{partial v_0}frac{dv_0}{dx}+frac{partial p}{partial v_1}frac{dv_1}{dx}$$
$endgroup$
add a comment |
$begingroup$
Note that $$p=p(v_0,v_1)\
frac{dp}{dx}=frac{partial p}{partial v_0}frac{dv_0}{dx}+frac{partial p}{partial v_1}frac{dv_1}{dx}$$
$endgroup$
add a comment |
$begingroup$
Note that $$p=p(v_0,v_1)\
frac{dp}{dx}=frac{partial p}{partial v_0}frac{dv_0}{dx}+frac{partial p}{partial v_1}frac{dv_1}{dx}$$
$endgroup$
Note that $$p=p(v_0,v_1)\
frac{dp}{dx}=frac{partial p}{partial v_0}frac{dv_0}{dx}+frac{partial p}{partial v_1}frac{dv_1}{dx}$$
answered Dec 13 '18 at 13:05
Empy2Empy2
33.5k12261
33.5k12261
add a comment |
add a comment |
$begingroup$
Your result is wrong because $p$ is a function of $x$ that contains the two functions $nu_0$ and $nu_1$ but not in the nested form $p(nu_0(nu_1))$. So we must write its derivative using also the quotient rule as:
$$
frac{partial p}{partial x}=frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
and the correct final result is
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
$endgroup$
add a comment |
$begingroup$
Your result is wrong because $p$ is a function of $x$ that contains the two functions $nu_0$ and $nu_1$ but not in the nested form $p(nu_0(nu_1))$. So we must write its derivative using also the quotient rule as:
$$
frac{partial p}{partial x}=frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
and the correct final result is
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
$endgroup$
add a comment |
$begingroup$
Your result is wrong because $p$ is a function of $x$ that contains the two functions $nu_0$ and $nu_1$ but not in the nested form $p(nu_0(nu_1))$. So we must write its derivative using also the quotient rule as:
$$
frac{partial p}{partial x}=frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
and the correct final result is
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
$endgroup$
Your result is wrong because $p$ is a function of $x$ that contains the two functions $nu_0$ and $nu_1$ but not in the nested form $p(nu_0(nu_1))$. So we must write its derivative using also the quotient rule as:
$$
frac{partial p}{partial x}=frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
and the correct final result is
$$
frac{partial L}{partial x} = frac{partial L}{partial p} cdot frac{e^{nu_0}frac{partial nu_0}{partial x}(e^{v_{0}}+e^{v_{1}})-e^{nu_0}(e^{v_{0}}frac{partial nu_0}{partial x}+e^{v_{1}}frac{partial nu_1}{partial x})}{(e^{v_{0}}+e^{v_{1}})^2}
$$
edited Dec 13 '18 at 13:10
answered Dec 13 '18 at 12:59
Emilio NovatiEmilio Novati
52k43474
52k43474
add a comment |
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%2f3037950%2fchain-rule-doubt%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