Differential under an integral sign.

[binbagsss]

Sorry I'm not sure how you get math script on here, so have had to attach it.

It is a differential under an integral sign and I'm not too sure how to approach it.
Would you use Leibniz rule?
Do you differentiate or integrate first?
- where y is a function of x, and R of y

Thanks in advance

 

[STEMucator]

You don't need to integrate here.

You have y as a function of x and R as a function of y. You want to find the partial of R with respect to y which you know is a function of x. So when you differentiate the integrand, don't forget to apply the chain rule.

 

[vanhees71]

No! What's asked for is the functional derivative. You get it by evaluating, how the integral changes when the function y(x) changes by an infinitesimal \delta y(x). Unfortunately the problem is not completely stated. You also need boundary conditions. If it's a usual variational problem like in Hamilton's principle, the values at the end points are fixed.