Proof Since f is differentiable at c, we have
( )
( )
lim
( )
x
c
f x
f c
f c
x
c
→
−
=
′
−
But for x ≠ c, we have
f (x) – f (c) =
( )
( ) (
)
f x
f c
x
c
x
c
−
−
−
Therefore
lim[ ( )
( )]
x
c f x
f c
→
−
 =
( )
( )
lim (
)
x
c
f x
f c
x
c
x
c
→
−


−


−


or
lim[ ( )]
lim[ ( )]
x
c
x
c
f x
f c
→
−→
 =
( )
( )
lim lim [(
)]
x
c
x
c
f x
f c
x
c
x
c
→
→
−


−


−


= f′(c)