1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Splitting the derivative?

  1. May 26, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex]\frac{de_{s}}{dT} = \frac{L_{v}e_{s}}{R_{v}T^{2}} [/itex]


    [itex] e_{s}(T) = 6.11 e^{\frac{L}{RV}(\frac{1}{T}-\frac{1}{273})} [/itex]

    2. Relevant equations

    3. The attempt at a solution

    The way my lecturer derived it was he 'split' the derivative and took them to their respective sides and integrated.

    So he got

    [itex] \frac{de_{s}}{e_{s}} = \frac{LdT}{R_{v}T^{2}} [/itex]

    However I was under the impression that you can't 'split' a derivative like that. Is this just a shortcut some physicists take to make the maths more simple? If it is what is the correct way of deriving this?
  2. jcsd
  3. May 26, 2012 #2


    User Avatar
    Homework Helper

  4. May 26, 2012 #3
    How accepted is it? I am doing a double degree in pure mathematics and physics. If I was to do this in my pure maths classes would I be stoned?
  5. May 26, 2012 #4
    I've taken calculus 1 and I've seen it before? :s
  6. May 26, 2012 #5
    Sorry I wasn't very clear. I have seen it before and done it countless times I just always have a memory of a teacher saying don't tell a pure mathematician about it.
  7. May 26, 2012 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    In my experience, physicists & engineers are notorious for treating Leibniz's notation for the derivative as if it were a fraction.

    A more rigorous handling of this derivative equation might be as follows:
    Treating es as a function of T we have

    [itex]\displaystyle \frac{d}{dT}(e_{s}) = \frac{L_{v}e_{s}}{R_{v}T^{2}}[/itex]

    Rewriting this equation gives us

    [itex]\displaystyle \frac{1}{e_{s}}\frac{d}{dT}(e_{s}) = \frac{L}{R_{v}T^{2}} [/itex]

    Integrating w.r.t. T gives

    [itex]\displaystyle \int {\frac{1}{e_{s}}\frac{d}{dT}(e_{s})}dT = \int{\frac{L}{R_{v}T^{2}}}dT [/itex]

    We can rewrite the integral on the left hand side.

    [itex]\displaystyle \int {\frac{1}{e_{s}}\frac{d}{dT}(e_{s})}dT = \int {\frac{1}{e_{s}}}\,de_{s}[/itex]​

    Alternatively, if [itex]\displaystyle \frac{d}{dT}(e_{s}) = \frac{L_{v}e_{s}}{R_{v}T^{2}}\,,[/itex]
    then the differential of es is given by [itex]\displaystyle d\,e_{s} = \frac{L_{v}e_{s}}{R_{v}T^{2}}\,dT[/itex]
  8. May 27, 2012 #7
    exactly what I wanted. thank you
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook