Generated by the Physics Derivation Graph. Eq. \ref{eq:1029383} is an initial equation. \begin{equation} y = \cos(x)+i \sin(x) \label{eq:1029383} \end{equation} Differentiate Eq. \ref{eq:1029383} with respect to \(x\); yields Eq. \ref{eq:1838300}. \begin{equation} \frac{d}{dx} y = -\sin(x) + i\cos(x) \label{eq:1838300} \end{equation} Factor \(i\) from the RHS of Eq. \ref{eq:1838300}; yields Eq. \ref{eq:2948271}. \begin{equation} \frac{d}{dx} y = (i\sin(x) + \cos(x)) i \label{eq:2948271} \end{equation} Substitute RHS of Eq. \ref{eq:1029383} into Eq. \ref{eq:2948271}; yields Eq. \ref{eq:9038289}. \begin{equation} \frac{d}{dx} y = y i \label{eq:9038289} \end{equation} Multiply both sides of Eq. \ref{eq:9038289} by \(dx\); yields Eq. \ref{eq:1111289}. \begin{equation} dy = y i dx \label{eq:1111289} \end{equation} Divide both sides of Eq. \ref{eq:1111289} by \(y\); yields Eq. \ref{eq:8883737}. \begin{equation} \frac{dy}{y} = i dx \label{eq:8883737} \end{equation} Indefinite integral of RHS of Eq. \ref{eq:8883737} over \(y\); yields Eq. \ref{eq:9984877}. \begin{equation} \log(y) = i dx \label{eq:9984877} \end{equation} Indefinite integral of RHS of Eq. \ref{eq:9984877} over \(x\); yields Eq. \ref{eq:3747585}. \begin{equation} \log(y) = i x \label{eq:3747585} \end{equation} Swap LHS of Eq. \ref{eq:3747585} with RHS; yields Eq. \ref{eq:3784785}. \begin{equation} i x = \log(y) \label{eq:3784785} \end{equation} Make Eq. \ref{eq:3784785} the power of \(e\); yields Eq. \ref{eq:9587572}. \begin{equation} \exp(i x) = y \label{eq:9587572} \end{equation} Substitute RHS of Eq. \ref{eq:9587572} into Eq. \ref{eq:1029383}; yields Eq. \ref{eq:8888888}. \begin{equation} \exp(i x) = \cos(x)+i \sin(x) \label{eq:8888888} \end{equation} Eq. \ref{eq:8888888} is one of the final equations.