If u(x) = f(x) + ig(x) is a complex-valued function of a real variable x and the real and imaginary parts f(x) and g(x) are differentiable functions of x, then the derivative of u is defined to be u'(x) = f '(x) + ig'(x). Use this together with ex + iy = x iy = x(cos(y) + i sin(y)) to prove that if F(x) = erx , then F '(x) = r x when f = a + bi is a complex number.