derivative of (tan(x)/(cos(x)))

解

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

- sec (x) + 2 sec^{3} (x)

解答ステップ

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

簡素化

\frac{tan ( x )}{cos ( x )} : sec (x) tan (x)

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sec (x), g = tan (x)

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

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

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

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

簡素化

sec (x) tan (x) tan (x) + sec^{2} (x) sec (x) : - sec (x) + 2 sec^{3} (x)

= - sec (x) + 2 sec^{3} (x)