Commutable Derivative and Integral in Multivariable Calculus

[matness]

it is simple but i have some suspession about it
when the integral and derivative of some func can commute ?
for ex. is it possible to say
$$\frac{{\partial ^{} }}{{\partial y^{} }}\int_a^b {f(x,y)dx} = \int_a^b {\frac{{\partial ^{} }}{{\partial y^{} }}f(x,y)dx}$$



or are there any condition for f(x,y) to satisfy?(?any toplogical condition other than f integrable)

 

[quasar987]

I my book, the theorem reads: "If M(x,y) and dM/dy are continuous functions on some region R, then < what you wrote >"

I assume it is implied that the region R contains the interval [a,b]

 

[matness]

thanks for reply
can you give me the name/author of the book or thm itself ?

 

[benorin]

The operation you described is called "differentiation under the integral sign."

Here is a link to a paper entitled Necessary and sufficient conditions for differentiating under the integral sign.

 

[quasar987]

Book has no name. Written by my college professor or multivariable calculus.