derivative of cos^3(x/(x+1))

解

\frac{d}{d x} (cos^{3} (\frac{x}{x + 1}))

- \frac{3 cos ^{2} ( \frac{x}{x + 1} ) sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (cos^{3} (\frac{x}{x + 1}))

連鎖法則を適用:

3 (cos (\frac{x}{x + 1}))^{2} \frac{d}{d x} (cos (\frac{x}{x + 1}))

= 3 (cos (\frac{x}{x + 1}))^{2} \frac{d}{d x} (cos (\frac{x}{x + 1}))

\frac{d}{d x} (cos (\frac{x}{x + 1})) = - \frac{sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}}

= 3 (cos (\frac{x}{x + 1}))^{2} (- \frac{sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}})

簡素化

3 cos^{2} (\frac{x}{x + 1}) (- \frac{sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}}) : - \frac{3 cos ^{2} ( \frac{x}{x + 1} ) sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}}

= - \frac{3 cos ^{2} ( \frac{x}{x + 1} ) sin ( \frac{x}{x + 1} )}{( x + 1 ) ^{2}}