For this problem the outside function is (hopefully) clearly the trig function and the inside function is the stuff inside of the trig function. The derivative is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{R'\left( w \right) = - 7\csc \left( {7w} \right)\cot \left( {7w} \right)}}\]

In dealing with functions like cosecant (or secant for that matter) be careful to make sure that the inside function gets substituted into both terms of the derivative of the outside function. One of the more common mistakes with this kind of problem is to only substitute the \(7w\) into only the cosecant or only the cotangent instead of both as it should be.