"Use the definition of derivative :"
f'(x) = lim_{h->0} (f(x+h) - f(x))/h
"Here we have"
f'(x_0) = lim_{h->0} (f(x_0 + h) - f(x_0))/h
g'(x_0) = lim_{h->0} (g(x_0 + h) - g(x_0))/h
"We need to prove that"
f'(x_0) = g'(x_0)
"or"
f'(x_0) - g'(x_0) = 0
"or"
h'(x_0) = 0
"with "h(x) = f(x) - g(x)
"or"
lim_{h->0} (f(x_0 + h) - g(x_0 +h) - f(x_0) + g(x_0))/h = 0
"or"
lim_{h->0} (f(x_0 + h) - g(x_0 + h))/h = 0
"(due to "f(x_0) = g(x_0)")"
"Now"
f(x_0 + h) <= g(x_0 + h)
=> lim <= 0 " if "h>0" and "lim >= 0" if "h < 0
"We made the assumption that f and g are differentiable"
"so "h(x) = f(x) - g(x)" is also differentiable,"
"so the left limit must be equal to the right limit, so"
=> lim = 0
=> h'(x_0) = 0
=> f'(x_0) = g'(x_0)
