sin x, cos x の微分

2014/4/10 2015/3/15 Learning, Mathematics

$\sin x$ 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

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

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

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

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

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

[TeX]

\begin{eqnarray}
\frac{d}{dx} \sin x &=& \lim_{\Delta x \to 0} \frac{\sin (x + \Delta x) - \sin x}{\Delta x} \\
                    &=& \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 \} \\
                    &=& \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 \} \\
                    &=& \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 \} \\
                    &=& \cos x  ~~~~\left ( \because \lim_{\Delta x \to 0} \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} = 1 \right )
\end{eqnarray}

\begin{eqnarray}
\frac{d}{dx} \cos x &=& \lim_{\Delta x \to 0} \frac{\cos (x + \Delta x) - \cos x}{\Delta x} \\
                    &=& \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 \} \\
                    &=& \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 \} \\
                    &=& \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 \} \\
                    &=& -\sin x  ~~~~\left ( \because \lim_{\Delta x \to 0} \frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}} = 1 \right )
\end{eqnarray}

\begin{eqnarray}

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

&=& \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 \} \\

&=& \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 \} \\

&=& \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 \} \\

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

\end{eqnarray}

\begin{eqnarray}

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

&=& \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 \} \\

&=& \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 \} \\