derivative of (x^{sin(x})/(cos(x)))

解

\frac{d}{d x} (\frac{x ^{s i n (x)}}{cos ( x )})

x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x}) sec (x) + sec (x) tan (x) x^{s i n (x)}

解答ステップ

\frac{d}{d x} (\frac{x ^{s i n (x)}}{cos ( x )})

簡素化

\frac{x ^{s i n (x)}}{cos ( x )} : x^{s i n (x)} sec (x)

= \frac{d}{d x} (x^{s i n (x)} sec (x))

積の法則を適用:

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

f = x^{s i n (x)}, g = sec (x)

= \frac{d}{d x} (x^{s i n (x)}) sec (x) + \frac{d}{d x} (sec (x)) x^{s i n (x)}

\frac{d}{d x} (x^{s i n (x)}) = x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x})

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

= x^{s i n (x)} (cos (x) ln (x) + \frac{sin ( x )}{x}) sec (x) + sec (x) tan (x) x^{s i n (x)}