# Quick question about the derivative of a complex-valued function

1. Feb 20, 2010

### tjackson3

This is something that has been bothering me for awhile. Suppose we have some function f(x) = u(x) + iv(x), where u,v are real-valued functions. Obviously Re(f) = u(x) and Im(f) = v(x), both of which are real. My question is: in general, is it safe to say that

$$\frac{d}{dx} Re(f) = Re\left(\frac{df}{dx}\right)$$

? If so, that makes life a lot easier with trig identities dealing with derivatives. This seems really trivial, and I feel like it's correct, but I can't figure out how one would prove it, and I don't want to use it just assuming that it is true...

Thanks!

2. Feb 20, 2010

### tiny-tim

Hi tjackson3!

no problem … df(x)/dx = du(x)/dx + idv(x)/dx,

and if u,v (and x) are real-valued functions, then so are du(x)/dx, dv(x)/dx.

3. Feb 21, 2010

### notmuch

tjackson3, is x a real variable?