Using Taylor expansion, show that
f0
(x0) = f(x0 + h) − f(x0)
h − h
2
f00(ξ),
for some ξ lying in between x0 and x0 + h.
Solution: We expand the function f in a first order Taylor polynomial around x0:
f(x) = f(x0)+(x − x0)f0
(x0)+(x − x0)
2 f00(ξ)
2 ,
where ξ is between x and x0. Let x = x0 + h:
f(x0 + h) = f(x0) + hf0
(x0) + h2
2 f00(ξ).
Solving for f0
(x0), we obtain:
f0
(x0) = f(x0 + h) − f(x0)
h − h
2
f00(ξ)