derivative of (x^x/(sin(x)))

解

\frac{d}{d x} (\frac{x ^{x}}{sin ( x )})

x^{x} csc (x) (ln (x) + 1) - x^{x} cot (x) csc (x)

解答ステップ

\frac{d}{d x} (\frac{x ^{x}}{sin ( x )})

簡素化

\frac{x ^{x}}{sin ( x )} : x^{x} csc (x)

= \frac{d}{d x} (x^{x} csc (x))

積の法則を適用:

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

f = x^{x}, g = csc (x)

= \frac{d}{d x} (x^{x}) csc (x) + \frac{d}{d x} (csc (x)) x^{x}

\frac{d}{d x} (x^{x}) = x^{x} (ln (x) + 1)

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

= x^{x} (ln (x) + 1) csc (x) + (- cot (x) csc (x)) x^{x}

簡素化

= x^{x} csc (x) (ln (x) + 1) - x^{x} cot (x) csc (x)