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