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

解

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

- \frac{1}{- 1 + sin ( x )}

解答ステップ

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

簡素化

\frac{1 + sin ( x )}{cos ( x )} : sec (x) + tan (x)

= \frac{d}{d x} (sec (x) + tan (x))

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (sec (x)) + \frac{d}{d x} (tan (x))

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

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

= sec (x) tan (x) + sec^{2} (x)

簡素化

sec (x) tan (x) + sec^{2} (x) : - \frac{1}{- 1 + sin ( x )}

= - \frac{1}{- 1 + sin ( x )}