derivative of ln((1+sin(4x)/(cos(4x))))

解

\frac{d}{d x} (ln (\frac{1 + sin ( 4 x )}{cos ( 4 x )}))

\frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( sin ( 4 x ) + 1 )}{cos ( 4 x ) ( sin ( 4 x ) + 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{1 + sin ( 4 x )}{cos ( 4 x )}))

連鎖法則を適用:

\frac{1}{\frac{1 + s i n ( 4 x )}{c o s ( 4 x )}} \frac{d}{d x} (\frac{1 + sin ( 4 x )}{cos ( 4 x )})

= \frac{1}{\frac{1 + s i n ( 4 x )}{c o s ( 4 x )}} \frac{d}{d x} (\frac{1 + sin ( 4 x )}{cos ( 4 x )})

\frac{d}{d x} (\frac{1 + sin ( 4 x )}{cos ( 4 x )}) = \frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( 1 + sin ( 4 x ) )}{cos ^{2} ( 4 x )}

= \frac{1}{\frac{1 + s i n ( 4 x )}{c o s ( 4 x )}} \cdot \frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( 1 + sin ( 4 x ) )}{cos ^{2} ( 4 x )}

簡素化

\frac{1}{\frac{1 + s i n ( 4 x )}{c o s ( 4 x )}} \cdot \frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( 1 + sin ( 4 x ) )}{cos ^{2} ( 4 x )} : \frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( sin ( 4 x ) + 1 )}{cos ( 4 x ) ( sin ( 4 x ) + 1 )}

= \frac{4 cos ^{2} ( 4 x ) + 4 sin ( 4 x ) ( sin ( 4 x ) + 1 )}{cos ( 4 x ) ( sin ( 4 x ) + 1 )}