sin x, cos x の微分

  • \sin x の微分

\displaystyle \frac{d}{dx} \sin x = \lim_{\Delta x \to 0} \frac{\sin (x + \Delta x) - \sin x}{\Delta x}

                \displaystyle = \lim_{\Delta x \to 0} \frac{1}{\Delta x} \cdot 2 \left \{ \cos \left (\frac{x + \Delta x + x}{2} \right) \sin \left( \frac{x + \Delta x - x}{2} \right) \right \}

                \displaystyle = \lim_{\Delta x \to 0} \frac{2}{\Delta x} \left \{ \cos \left ( \frac{2x + \Delta x}{2} \right ) \sin \left( \frac{\Delta x}{2} \right ) \right \}

                \displaystyle = \lim_{\Delta x \to 0} \left \{\cos \left ( \frac{2x + \Delta x}{2} \right ) \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} \right \}

                \displaystyle = \cos x ~~~~\left ( \because \lim_{\Delta x \to 0} \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} = 1 \right )

  • \cos x の微分

\displaystyle \frac{d}{dx} \cos x = \lim_{\Delta x \to 0} \frac{\cos (x + \Delta x) - \cos x}{\Delta x}

                \displaystyle = \lim_{\Delta x \to 0} \frac{1}{\Delta x} \cdot (-2) \left \{ \sin \left (\frac{x + \Delta x + x}{2} \right) \sin \left( \frac{x + \Delta x - x}{2} \right) \right \}

                \displaystyle = \lim_{\Delta x \to 0} \frac{-2}{\Delta x} \left \{ \sin \left ( \frac{2x + \Delta x}{2} \right ) \sin \left( \frac{\Delta x}{2} \right ) \right \}

                \displaystyle = \lim_{\Delta x \to 0} \left \{-\sin \left ( \frac{2x + \Delta x}{2} \right ) \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} \right \}

                \displaystyle = -\sin x  ~~~~\left ( \because \lim_{\Delta x \to 0} \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} = 1 \right )

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