MHB How Is the Derivative of e^(i*theta) Derived?

[E01]

I have a fairly simple question. My teacher used the fact that $\Delta e^{i\theta} = e^{i\theta}i\Delta \theta$ when theta approaches 0. Does he derive this using the fact that the derivate of $e^t = te^t$ ?

 

[Prove It]

I have a fairly simple question. My teacher used the fact that $\Delta e^{i\theta} = e^{i\theta}i\Delta \theta$ when theta approaches 0. Does he derive this using the fact that the derivate of $e^t = te^t$ ?

Well, first of all, $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \left( \mathrm{e}^t \right) = \mathrm{e}^t \end{align*}$, NOT $\displaystyle \begin{align*} t\,\mathrm{e}^t \end{align*}$.

What your teacher has done is use the Chain Rule.