Derivative to function ratio

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Is there a physical meaning for derivative of the function to function ratio? That is, this quantity,
$$
Q(x) = frac1f(x)fracdf(x)dx
$$
Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio.
Or even what can we say about $Q(x)$, say when $Q(x)<0$.




derivatives




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  • This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
    – Nick
    15 hours ago






  • 1




    I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
    – user2167741
    15 hours ago














up vote
1
down vote

favorite
1












Is there a physical meaning for derivative of the function to function ratio? That is, this quantity,
$$
Q(x) = frac1f(x)fracdf(x)dx
$$
Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio.
Or even what can we say about $Q(x)$, say when $Q(x)<0$.




derivatives




share|cite|improve this question



















  • This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
    – Nick
    15 hours ago






  • 1




    I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
    – user2167741
    15 hours ago












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Is there a physical meaning for derivative of the function to function ratio? That is, this quantity,
$$
Q(x) = frac1f(x)fracdf(x)dx
$$
Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio.
Or even what can we say about $Q(x)$, say when $Q(x)<0$.




derivatives




share|cite|improve this question











Is there a physical meaning for derivative of the function to function ratio? That is, this quantity,
$$
Q(x) = frac1f(x)fracdf(x)dx
$$
Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio.
Or even what can we say about $Q(x)$, say when $Q(x)<0$.





derivatives





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share|cite|improve this question




share|cite|improve this question









asked 16 hours ago









user2167741

979




979











  • This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
    – Nick
    15 hours ago






  • 1




    I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
    – user2167741
    15 hours ago
















  • This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
    – Nick
    15 hours ago






  • 1




    I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
    – user2167741
    15 hours ago















This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
– Nick
15 hours ago




This is the scaling factor of the derivitive to the function. This quantity gives an intuition of how the function relates to its gradient.
– Nick
15 hours ago




1




1




I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
– user2167741
15 hours ago




I also learned that this is called 'logarithmic derivative'. en.wikipedia.org/wiki/Logarithmic_derivative
– user2167741
15 hours ago










4 Answers
4






active

oldest

votes

















up vote
1
down vote













It is also $frac d(ln(f(x))dx$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.






share|cite|improve this answer




























    up vote
    1
    down vote













    This would give
    $$
    intlimits_x_0^x !Q(xi) , dxi = ln fracf(x)f(x_0) iff \
    f(x) = f(x_0) exp intlimits_x_0^x ! Q(xi) , dxi
    $$
    An example would be a radioactive decay process ($Q$: decay constant)



    Another: light absorption ($Q$: attenuation coefficient).






    share|cite|improve this answer






























      up vote
      1
      down vote













      I know in Physics they use that $fracdNdt=Nk$
      for various uses so you could use it to show whether a relationship is exponential or not






      share|cite|improve this answer





















      • I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
        – user2167741
        15 hours ago






      • 1




        Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
        – Henry Lee
        15 hours ago

















      up vote
      1
      down vote













      If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.



      For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) , e^rt$.



      But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.






      share|cite|improve this answer





















      • Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
        – user2167741
        12 hours ago










      • You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
        – Hans Lundmark
        12 hours ago










      • No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
        – user2167741
        11 hours ago










      • Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
        – Hans Lundmark
        11 hours ago











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      4 Answers
      4






      active

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      4 Answers
      4






      active

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      active

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      active

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      up vote
      1
      down vote













      It is also $frac d(ln(f(x))dx$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.






      share|cite|improve this answer

























        up vote
        1
        down vote













        It is also $frac d(ln(f(x))dx$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.






        share|cite|improve this answer























          up vote
          1
          down vote










          up vote
          1
          down vote









          It is also $frac d(ln(f(x))dx$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.






          share|cite|improve this answer













          It is also $frac d(ln(f(x))dx$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered 15 hours ago









          Ross Millikan

          275k21183348




          275k21183348




















              up vote
              1
              down vote













              This would give
              $$
              intlimits_x_0^x !Q(xi) , dxi = ln fracf(x)f(x_0) iff \
              f(x) = f(x_0) exp intlimits_x_0^x ! Q(xi) , dxi
              $$
              An example would be a radioactive decay process ($Q$: decay constant)



              Another: light absorption ($Q$: attenuation coefficient).






              share|cite|improve this answer



























                up vote
                1
                down vote













                This would give
                $$
                intlimits_x_0^x !Q(xi) , dxi = ln fracf(x)f(x_0) iff \
                f(x) = f(x_0) exp intlimits_x_0^x ! Q(xi) , dxi
                $$
                An example would be a radioactive decay process ($Q$: decay constant)



                Another: light absorption ($Q$: attenuation coefficient).






                share|cite|improve this answer

























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  This would give
                  $$
                  intlimits_x_0^x !Q(xi) , dxi = ln fracf(x)f(x_0) iff \
                  f(x) = f(x_0) exp intlimits_x_0^x ! Q(xi) , dxi
                  $$
                  An example would be a radioactive decay process ($Q$: decay constant)



                  Another: light absorption ($Q$: attenuation coefficient).






                  share|cite|improve this answer















                  This would give
                  $$
                  intlimits_x_0^x !Q(xi) , dxi = ln fracf(x)f(x_0) iff \
                  f(x) = f(x_0) exp intlimits_x_0^x ! Q(xi) , dxi
                  $$
                  An example would be a radioactive decay process ($Q$: decay constant)



                  Another: light absorption ($Q$: attenuation coefficient).







                  share|cite|improve this answer















                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 15 hours ago


























                  answered 15 hours ago









                  mvw

                  30.1k22150




                  30.1k22150




















                      up vote
                      1
                      down vote













                      I know in Physics they use that $fracdNdt=Nk$
                      for various uses so you could use it to show whether a relationship is exponential or not






                      share|cite|improve this answer





















                      • I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                        – user2167741
                        15 hours ago






                      • 1




                        Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                        – Henry Lee
                        15 hours ago














                      up vote
                      1
                      down vote













                      I know in Physics they use that $fracdNdt=Nk$
                      for various uses so you could use it to show whether a relationship is exponential or not






                      share|cite|improve this answer





















                      • I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                        – user2167741
                        15 hours ago






                      • 1




                        Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                        – Henry Lee
                        15 hours ago












                      up vote
                      1
                      down vote










                      up vote
                      1
                      down vote









                      I know in Physics they use that $fracdNdt=Nk$
                      for various uses so you could use it to show whether a relationship is exponential or not






                      share|cite|improve this answer













                      I know in Physics they use that $fracdNdt=Nk$
                      for various uses so you could use it to show whether a relationship is exponential or not







                      share|cite|improve this answer













                      share|cite|improve this answer



                      share|cite|improve this answer











                      answered 15 hours ago









                      Henry Lee

                      48210




                      48210











                      • I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                        – user2167741
                        15 hours ago






                      • 1




                        Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                        – Henry Lee
                        15 hours ago
















                      • I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                        – user2167741
                        15 hours ago






                      • 1




                        Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                        – Henry Lee
                        15 hours ago















                      I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                      – user2167741
                      15 hours ago




                      I get that - I know the relationship is exponential but wanted to know the physical significance of it. Like mvw says, attenuation coefficient gives a good meaning to it.
                      – user2167741
                      15 hours ago




                      1




                      1




                      Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                      – Henry Lee
                      15 hours ago




                      Hmmm I think it is quite hard to give a general meaning to something like that, you could also see it as some kind of relative stability.
                      – Henry Lee
                      15 hours ago










                      up vote
                      1
                      down vote













                      If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.



                      For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) , e^rt$.



                      But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.






                      share|cite|improve this answer





















                      • Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                        – user2167741
                        12 hours ago










                      • You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                        – Hans Lundmark
                        12 hours ago










                      • No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                        – user2167741
                        11 hours ago










                      • Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                        – Hans Lundmark
                        11 hours ago















                      up vote
                      1
                      down vote













                      If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.



                      For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) , e^rt$.



                      But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.






                      share|cite|improve this answer





















                      • Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                        – user2167741
                        12 hours ago










                      • You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                        – Hans Lundmark
                        12 hours ago










                      • No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                        – user2167741
                        11 hours ago










                      • Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                        – Hans Lundmark
                        11 hours ago













                      up vote
                      1
                      down vote










                      up vote
                      1
                      down vote









                      If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.



                      For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) , e^rt$.



                      But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.






                      share|cite|improve this answer













                      If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.



                      For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) , e^rt$.



                      But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.







                      share|cite|improve this answer













                      share|cite|improve this answer



                      share|cite|improve this answer











                      answered 13 hours ago









                      Hans Lundmark

                      32.7k563109




                      32.7k563109











                      • Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                        – user2167741
                        12 hours ago










                      • You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                        – Hans Lundmark
                        12 hours ago










                      • No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                        – user2167741
                        11 hours ago










                      • Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                        – Hans Lundmark
                        11 hours ago

















                      • Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                        – user2167741
                        12 hours ago










                      • You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                        – Hans Lundmark
                        12 hours ago










                      • No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                        – user2167741
                        11 hours ago










                      • Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                        – Hans Lundmark
                        11 hours ago
















                      Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                      – user2167741
                      12 hours ago




                      Thanks! Yes, I can see that in case of basic exponential/logistic model. What if $f(t)$ is the rate $r(t)$, where the growth rate $r$ depends on time? How do we interpret it?
                      – user2167741
                      12 hours ago












                      You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                      – Hans Lundmark
                      12 hours ago




                      You mean $Q(t)$? Then you get something which I don't know a name for, but which can be solved by separation of variables: $f'/f=r$ gives $int df/f=int r(t) , dt$, etc.
                      – Hans Lundmark
                      12 hours ago












                      No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                      – user2167741
                      11 hours ago




                      No, then I mean the quantity $r'(t)/r(t)$. I know $r$ follows exponential as well, but I don't know what the quantity would mean. It is some sort of relative rate of change of growth over time.
                      – user2167741
                      11 hours ago












                      Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                      – Hans Lundmark
                      11 hours ago





                      Well, if $r(t)$ is the per capita growth rate, then $r'(t)/r(t)$ is rate of change of the per capita growth rate per per capita growth rate... :-)
                      – Hans Lundmark
                      11 hours ago













                       

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