How to differentiate, for example, f(r,t ) wrt r without knowing in what way f depends on r and t? The 2019 Stack Overflow Developer Survey Results Are InFinding the Derivative Of $f(x) = 7ln(5xe^-x)$Chain rule and matrices - I'm confusedWirtinger derivative of composition of functionsNotation regarding different derivativesImplicit Second Derivatives using Partial Derivativeswhat is $fracdd(ax)f(x)$ and $fracdd(ax)f(x)$Chain rule for equations of multiple variablesDifferentiation of partial derivativesHow to compute this derivative of a square root of a sum?Partial derivative of a definite integral
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How to differentiate, for example, f(r,t ) wrt r without knowing in what way f depends on r and t?
The 2019 Stack Overflow Developer Survey Results Are InFinding the Derivative Of $f(x) = 7ln(5xe^-x)$Chain rule and matrices - I'm confusedWirtinger derivative of composition of functionsNotation regarding different derivativesImplicit Second Derivatives using Partial Derivativeswhat is $fracdd(ax)f(x)$ and $fracdd(ax)f(x)$Chain rule for equations of multiple variablesDifferentiation of partial derivativesHow to compute this derivative of a square root of a sum?Partial derivative of a definite integral
$begingroup$
Say we have some function $psi'(r,t)$, given by $psi'(r,t)=e^af(r,t)psi(r,t)$
and we want to calculate $nabla^2psi'(r,t)$.
I obviously know how to do all the basic steps of this product rule , chain rule and all that but there's one part I'm a little unsure of ( I always found it a little confusing , I used to know how to do it but I forgot )
Basically in the parts of the equation where we have $tfracpartial f(r,t)partial r,tfracpartial f(r,t) partial t, tfracpartial psi(r,t)partial r,tfracpartialpsi(r,t)partial t$, I want to know how we can deal with these ?
(If what I'm asking isn't clear , please ask me to elucidate and I'll write more about what it is specifically that confuses me )
derivatives implicit-differentiation
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
$endgroup$
add a comment |
$begingroup$
Say we have some function $psi'(r,t)$, given by $psi'(r,t)=e^af(r,t)psi(r,t)$
and we want to calculate $nabla^2psi'(r,t)$.
I obviously know how to do all the basic steps of this product rule , chain rule and all that but there's one part I'm a little unsure of ( I always found it a little confusing , I used to know how to do it but I forgot )
Basically in the parts of the equation where we have $tfracpartial f(r,t)partial r,tfracpartial f(r,t) partial t, tfracpartial psi(r,t)partial r,tfracpartialpsi(r,t)partial t$, I want to know how we can deal with these ?
(If what I'm asking isn't clear , please ask me to elucidate and I'll write more about what it is specifically that confuses me )
derivatives implicit-differentiation
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
$endgroup$
$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25
add a comment |
$begingroup$
Say we have some function $psi'(r,t)$, given by $psi'(r,t)=e^af(r,t)psi(r,t)$
and we want to calculate $nabla^2psi'(r,t)$.
I obviously know how to do all the basic steps of this product rule , chain rule and all that but there's one part I'm a little unsure of ( I always found it a little confusing , I used to know how to do it but I forgot )
Basically in the parts of the equation where we have $tfracpartial f(r,t)partial r,tfracpartial f(r,t) partial t, tfracpartial psi(r,t)partial r,tfracpartialpsi(r,t)partial t$, I want to know how we can deal with these ?
(If what I'm asking isn't clear , please ask me to elucidate and I'll write more about what it is specifically that confuses me )
derivatives implicit-differentiation
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
$endgroup$
Say we have some function $psi'(r,t)$, given by $psi'(r,t)=e^af(r,t)psi(r,t)$
and we want to calculate $nabla^2psi'(r,t)$.
I obviously know how to do all the basic steps of this product rule , chain rule and all that but there's one part I'm a little unsure of ( I always found it a little confusing , I used to know how to do it but I forgot )
Basically in the parts of the equation where we have $tfracpartial f(r,t)partial r,tfracpartial f(r,t) partial t, tfracpartial psi(r,t)partial r,tfracpartialpsi(r,t)partial t$, I want to know how we can deal with these ?
(If what I'm asking isn't clear , please ask me to elucidate and I'll write more about what it is specifically that confuses me )
derivatives implicit-differentiation
derivatives implicit-differentiation
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
asked Mar 30 at 17:23
can'tcauchycan'tcauchy
1,022417
1,022417
$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25
add a comment |
$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25
$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
WLOG, we can parameterize r and t by some parameter $tau$. We then have $$fracd fd tau = fracpartial fpartial rfracdrdtau+fracpartial fpartial tfracdtdtau$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.
answered Apr 1 at 10:37
Polly M.Polly M.
1
$endgroup$
add a comment |
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1 Answer
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$begingroup$
WLOG, we can parameterize r and t by some parameter $tau$. We then have $$fracd fd tau = fracpartial fpartial rfracdrdtau+fracpartial fpartial tfracdtdtau$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.
answered Apr 1 at 10:37
Polly M.Polly M.
1
$endgroup$
add a comment |
$begingroup$
WLOG, we can parameterize r and t by some parameter $tau$. We then have $$fracd fd tau = fracpartial fpartial rfracdrdtau+fracpartial fpartial tfracdtdtau$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.
answered Apr 1 at 10:37
Polly M.Polly M.
1
$endgroup$
add a comment |
$begingroup$
WLOG, we can parameterize r and t by some parameter $tau$. We then have $$fracd fd tau = fracpartial fpartial rfracdrdtau+fracpartial fpartial tfracdtdtau$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.
answered Apr 1 at 10:37
Polly M.Polly M.
1
$endgroup$
WLOG, we can parameterize r and t by some parameter $tau$. We then have $$fracd fd tau = fracpartial fpartial rfracdrdtau+fracpartial fpartial tfracdtdtau$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.
answered Apr 1 at 10:37
Polly M.Polly M.
1
answered Apr 1 at 10:37
Polly M.Polly M.
1
answered Apr 1 at 10:37
Polly M.Polly M.
1
answered Apr 1 at 10:37
Polly M.Polly M.
1
1
add a comment |
add a comment |
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$begingroup$
If you know what $f$ and $psi$ are, then you can evaluate those derivatives; otherwise, you just leave them like that. Or are you asking how to evaluate the partial derivatives of a multi-variable function?
$endgroup$
– Sambo
Mar 30 at 17:27
$begingroup$
@Sambo I was just asking is there a way to reduce them further than what I wrote above if we don't know anything about the form of the function except that they depend on r and t
$endgroup$
– can'tcauchy
Mar 30 at 17:32
$begingroup$
Then no, there is no way to reduce them further. Similar to how you can't reduce $df(x)/dx$ further.
$endgroup$
– Sambo
Mar 31 at 4:25