#"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)#