$$\frac{d}{dt}e^{it}$$

When taking the derivative of a \emph{complex-valued exponential}, we insist on the same property as a \emph{real-valued exponential}:
$$\frac{d}{dt}e^{\omega t} = \omega e^{\omega t} $$
$$\frac{d}{dt}e^{it} = i \cdot e^{it} e^{i\frac{\pi}{2}} \cdot e^{it} = e^{i(t + \frac{\pi}{2})}$$

\begin{nicebox}
$z \cdot i$ is rotation by $90^o$
\end{nicebox}

$$ z \cdot i = e^{i\theta} \cdot i = e^{i\theta} \cdot e^{i\frac{\pi}{2}} = e^{i(\theta + \frac{\pi}{2})}$$ 
$$ (\alpha + i\beta)i = i\alpha + i^2\beta = -\beta + i\alpha $$


% \includegraphics{images/euler/multrot.ps}
% \caption{ Multiplication and Rotation }

\begin{center}
\begin{pspicture}(-4,-4)(4,4)
		
% y-axis
\rput(0.3,3.75){ $Im$ }
\psline{->}(0,-3.75)(0,3.75)

% x-axis
\rput(3.75,0.3){ $Re$ }
\psline{->}(-3.75,0)(3.75,0)
			
% the circle
\pscircle(0,0){ 3 }
			
% quadrants
\rput(0.3,3.3){ $i^1$ }
\rput(0.3,-3.3){ $i^3$ }
\rput(-3.3,0.3){ $i^2$ }
\rput(3.3,0.3){ $i^0$ }
	
 % new vector

\psline[linewidth=1.5 pt]{->}(0,0)(2.598,1.500)
\rput(2,1.500){$\alpha+i\beta$}

%\pcline[linewidth=1.5 pt]{->}(0,0)(2.598,1.500)
%\Aput{\small $\alpha+i\beta$}

 % new vector
\psline[linewidth=1.5 pt]{->}(0,0)(-1.500,2.598)
\rput(-1.500,1.5){$-\beta+i\alpha$}

%\pcline[linewidth=1.5 pt]{->}(0,0)(-1.500,2.598)
%\Aput{\small $-\beta+i\alpha$}
	
\end{pspicture}
\end{center}

Derivatives of Complex Exponentials