Euler's equation

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\[\begin{aligned} &\text { Let } f(x)=\frac{\cos x+i \sin x}{e^{i x}}\\\\ &\Rightarrow f^{\prime}(x)=\frac{\left(i \cos x-\sin x) e^{i x}-i e^{i x}(\cos x+i \sin x)\right.}{e^{2 i x}}\\\\ &=\frac{e^{i x}(i \cos x-\sin x-i \cos x+\sin x)}{e^{2 i x}}\\\\ &f^{\prime}(x)=0\\\\ &\Rightarrow f \text { is constant }\\\\ &\text { then } f(x)=f(0)\\\\ &\Leftrightarrow \frac{\cos x+i \sin x}{e^{i x}}=1\\\\ &\text { thus } e^{i x}=\cos x+i \sin x \end{aligned}\]

\[e^{i x}=\cos x+i \sin x \]

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