The derivative of sin (x) is cos (x). It does not work the other way around, though. The derivative of cos (x) is -sin (x).

Rarely. Projecting a movie, with no alteration at all, would be public performance; re-cutting the film would be derivative. An example that is both public performance and derivative would be a cover performance of a song, done in an entirely different style. Both public performance and creation of a derivative work require permission from the copyright holder.

A photograph of a statue would be considered a derivative work; if the statue is protected by copyright, you would need permission from the rightsholder to create the derivative work.

"Derivative of"

well, the second derivative is the derivative of the first derivative. so, the 2nd derivative of a function's indefinite integral is the derivative of the derivative of the function's indefinite integral. the derivative of a function's indefinite integral is the function, so the 2nd derivative of a function's indefinite integral is the derivative of the function.