Can the derivative of a real function be imaginary?

No. The derivative is defined as $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ This is a limit of real numbers, hence if it exists it is real.

No, it cannot. Since $f'(x_0)=\lim_{x\to x_0}\dfrac{f(x)-f(x_0)}{x-x_0}$ and since the limit at a point of a real function is (if it exists) a real number, $f'(x_0)\in\mathbb R$.