How would we differentiate $\cos(z)$ or $\sinh(z)$ for example?

I'm currently taking a complex analysis and it seems like we went straight from the definition of complex derivative straight into just assuming that the derivatives behave at lot like they do for real variables.

My question is how do we know that $\frac{d(\cos(z))}{dz}=-\sin(z)$ and similarly for other trig functions and the exponential function.

Is there something more difficult going on or is it just the application of the definition?

I tried writing $\cos(z)$ as $\cos(x+iy)$ but I don't even really know what the definition of $\cos(z)$ actually is.

I also tried the exponential form approach but realised I don't know how to differentiate $e^{iz}$ either.

Can anyone shed some light on this and maybe provide a few examples?

Thanks.

• Define $e^{x+iy}=e^x(\cos y+i\sin y)$. Verify that it satisfies the Cauchy-Riemann equations. Its complex derivative is the derivative with respect to $x$. The chain rule also applies. – Matt Samuel Mar 3 '16 at 0:26

You know that $\sin$ is defined as $$\sin z = \frac{1}{2i}(e^{i z} - e^{-iz})$$ so it remains to prove that $\frac{d\exp}{dz} = \exp$ and the chain rule also works. The former is a simple practice of power series.