$\tan x$ の微分
\begin{eqnarray*} \tan{x} &=& \frac{\sin{x}}{\cos{x}}\\ \tan 'x &=& \frac{\sin 'x \cos x - \sin x \cos 'x}{\cos^{2} x}\\ &=& \frac{\cos^{2} x + \sin^{2} x}{\cos^{2} x}\\ &=& \frac{1}{\cos^{2} x} \end{eqnarray*}
\begin{eqnarray*} \tan{x} &=& \frac{\sin{x}}{\cos{x}}\\ \tan 'x &=& \frac{\sin 'x \cos x - \sin x \cos 'x}{\cos^{2} x}\\ &=& \frac{\cos^{2} x + \sin^{2} x}{\cos^{2} x}\\ &=& \frac{1}{\cos^{2} x} \end{eqnarray*}