Consider
We will now employa methodthat will expressthe right side of this equationas a Riemann
Sum.
In particular,
Notethat is continuous.Hence,by the MeanValue Theorem,thereexist
suchthat
Hence
Takingthe limit of eachside as we have
We note that the left side is a constantand the right side is our definitionfor.
Hence
Proofof Theorem4.2
Let
By the MeanValue Theoremfor derivatives,thereexists suchthat