derivative arcsec(x)*arccos(x)

解

導関数

arcsec (x) \cdot arccos (x)

\frac{arccos ( x )}{x ^{2} x ^{2} - 1} - \frac{arcsec ( x )}{1 - x ^{2}}

解答ステップ

\frac{d}{d x} (arcsec (x) arccos (x))

積の法則を適用:

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

f = arcsec (x), g = arccos (x)

= \frac{d}{d x} (arcsec (x)) arccos (x) + \frac{d}{d x} (arccos (x)) arcsec (x)

\frac{d}{d x} (arcsec (x)) = \frac{1}{x ^{2} x ^{2} - 1}

\frac{d}{d x} (arccos (x)) = - \frac{1}{1 - x ^{2}}

= \frac{1}{x ^{2} x ^{2} - 1} arccos (x) + (- \frac{1}{1 - x ^{2}}) arcsec (x)

簡素化

\frac{1}{x ^{2} x ^{2} - 1} arccos (x) + (- \frac{1}{1 - x ^{2}}) arcsec (x) : \frac{arccos ( x )}{x ^{2} x ^{2} - 1} - \frac{arcsec ( x )}{1 - x ^{2}}

= \frac{arccos ( x )}{x ^{2} x ^{2} - 1} - \frac{arcsec ( x )}{1 - x ^{2}}