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 function for x, then the derivative of u is defined to be u'(x) = f'(x) + ig'(x).
Use this together with eˣ⁺ᶦʸ = eˣeᶦʸ = eˣ(cos(y) + i(sin(y)) to prove that if F(x)=eʳˣ, then F'(x)=eʳˣ when r = a+bi is a complex number