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Changing derivative into partial

  1. Sep 20, 2009 #1
    How do I know when it is ok to change a derivative into a partial derivative? For example there is something like:
    (d/dt)∫f(x,t)dx
    then they simplify it to:
    ∫[∂f(x,t)/∂t]dx
    I mean, it sort of makes sense to me, because the integral will be a function of t only so the (d/dt) is fine, but when you bring it inside the integral f(x,t) is a function of both t and x, so a partial is needed... but why is it ok to do this?
     
  2. jcsd
  3. Sep 20, 2009 #2

    CompuChip

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    When x is just an integration variable which does not depend on t, it is definitely allowed. In that case, namely,
    ∂f(x,t)/∂t = df(x,t)/dt.
     
  4. Sep 20, 2009 #3

    lurflurf

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    Several things
    -worry more about if the derivative can be interchanged with the integral to star
    -think about it the other way round
    if
    (∂/∂t)∫f(x,t)dx=∫[∂f(x,t)/∂t]dx
    does not bother you we can see changing d/dt to ∂/∂t is valid as x is a dummy variable
    -form the Newton Quotient to see this
     
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