derivative of (cos(x)/(csc(x)))

解

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

cos (2 x)

解答ステップ

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

簡素化

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

= \frac{d}{d x} (cos (x) sin (x))

積の法則を適用:

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

f = cos (x), g = sin (x)

= \frac{d}{d x} (cos (x)) sin (x) + \frac{d}{d x} (sin (x)) cos (x)

\frac{d}{d x} (cos (x)) = - sin (x)

\frac{d}{d x} (sin (x)) = cos (x)

= (- sin (x)) sin (x) + cos (x) cos (x)

簡素化

(- sin (x)) sin (x) + cos (x) cos (x) : cos (2 x)

= cos (2 x)