In the expression
Derivatives of compositions involving differentiable functions can always be found using the chain rule.
The composition of a function on itself, such as f.f, is customarily written f 2. (f.f)(x)=f(f(x))=f2(x). Likewise, (f.f.f)(x)=f(f(f(x)))=f3(x). By extension of this notation, f -1(x) is the inverse function of f.
However, for historical reasons, this superscript notation does not mean the same thing for trigonometric functions unless the superscript is negative: sin2(x) is shorthand for sin(x).sin(x).
In some cases, an expression for f in g(x)=f r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration.