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*))=*f*^{2}(*x*). Likewise, (*f*.*f*.*f*)(*x*)=*f*(*f*(*f*(*x*)))=*f*^{3}(*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: sin^{2}(*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.

See also: