derivative of ln((sin(x)/(cos(x))))

解

\frac{d}{d x} (ln (\frac{sin ( x )}{cos ( x )}))

2 csc (2 x)

解答ステップ

\frac{d}{d x} (ln (\frac{sin ( x )}{cos ( x )}))

簡素化

ln (\frac{sin ( x )}{cos ( x )}) : ln (tan (x))

= \frac{d}{d x} (ln (tan (x)))

連鎖法則を適用:

\frac{1}{tan ( x )} \frac{d}{d x} (tan (x))

= \frac{1}{tan ( x )} \frac{d}{d x} (tan (x))

\frac{d}{d x} (tan (x)) = sec^{2} (x)

= \frac{1}{tan ( x )} sec^{2} (x)

簡素化

\frac{1}{tan ( x )} sec^{2} (x) : 2 csc (2 x)

= 2 csc (2 x)