TITLE: Can the derivative of a real function be imaginary?
QUESTION [9 upvotes]: Suppose $f$ is a real function (e.g. mapping $\mathbb{R} \rightarrow \mathbb{R}$). Is it possible for its derivative (i.e. $\frac{d}{dx}f(x)$) to be imaginary?

REPLY [21 votes]: 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.

REPLY [4 votes]: 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, $f'(x_0)\in\mathbb R$.