The derivative of the complex exponential function \( e^{i\theta} \) is derived using the Chain Rule, resulting in the expression \( \Delta e^{i\theta} = e^{i\theta} i \Delta \theta \) as \( \Delta \theta \) approaches 0. The correct derivative of \( e^t \) is \( \frac{d}{dt}(e^t) = e^t \), not \( t e^t \). This clarification is essential for understanding how to apply the Chain Rule in the context of complex functions.
PREREQUISITESStudents of calculus, mathematicians, and anyone interested in the application of derivatives in complex analysis.
E01 said: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 $e^t = te^t$ ?