#"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)#