CK12 Calculus - Single Variable

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