Proof
We consider the auxiliary function
-
This function is also
continuous
and
differentiable
in
. Moreover, we have
and
-
![{\displaystyle {}g(b)=f(b)-(f(b)-f(a))=f(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74374181aa700d55284d5ef9d3c669884534954f)
Hence,
fulfills the conditions of
fact,
and therefore there exists some
,
such that
.
Because of the rules for derivatives, we obtain
-
![{\displaystyle {}f'(c)={\frac {f(b)-f(a)}{b-a}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e84c040106b21c321f6273194c7701017646796)