First we need a simple theorem.

Theorem:Suppose is a function defined on an open interval containing (except possibly at ) so that where is a real number. Suppose is a function defined on an open interval containing so that is continous at then is defined in an open interval containing (except possibly at )and.

Now we define a derivative.

Definition:Suppose is a function defined on an open interval containing . Define the function as then is defined on an open interval containing (except at ). If it happens that exists (real number) then we say isdifferenciableat and we call this limit .

Theorem 1:Suppose that is differenciable at defined on an open interval containing . Define the function for all such that then clearly is defined on an open interval containing (except at at )and.

Proof:The part where is defined around is straightforward. Now to show we use the theorem above. Let and let then the conditions of the theorem are satisfiedand.

Theorem 2:Suppose that is differenciable at defined on an open interval containing . Define the function for all such that then clearly is defined on an open interval containing (except at )and.

Proof:The proof is nearly identical to Theorem 1.

Corollarly:Suppose that is differenciable at defined on an open interval containing . Define the function for all such that then clearly is defined on an open interval containing (except at )and.

Proof:Note that . This means by the above theorems.