sin xの微分

$ f'(x) = \lim_{\Delta h \to 0}{f(x+\Delta h) - f(x) \over \Delta h}

$ \sin'(x) = \lim_{\Delta h \to 0}{\sin(x+\Delta h) - \sin(x) \over \Delta h}

$ \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta

$ \sin'(x) = \lim_{\Delta h \to 0}{\sin(x)\cos(\Delta h) + \cos(x)\sin(\Delta h) - \sin(x) \over \Delta h}

$ = \lim_{\Delta h \to 0}{\sin(x)\cos(\Delta h) + \cos(x)\sin(\Delta h) - \sin(x) \over \Delta h}

$ = \lim_{\Delta h \to 0}{\Bigg(\sin(x)\frac{\cos(\Delta h)-1}{\Delta h} + \cos(x)\frac{\sin(\Delta h)}{\Delta h}\Bigg)}