Changing derivative into partial

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?
 

CompuChip

Science Advisor
Homework Helper
4,284
47
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.
 

lurflurf

Homework Helper
2,417
122
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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