derivative of (sqrt(6x-x^2)/(tan(x)))

解

\frac{d}{d x} (\frac{6 x - x ^{2}}{tan ( x )})

- csc^{2} (x) 6 x - x^{2} + \frac{cot ( x ) ( 3 - x )}{6 x - x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{6 x - x ^{2}}{tan ( x )})

簡素化

\frac{6 x - x ^{2}}{tan ( x )} : cot (x) 6 x - x^{2}

= \frac{d}{d x} (cot (x) 6 x - x^{2})

積の法則を適用:

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

f = cot (x), g = 6 x - x^{2}

= \frac{d}{d x} (cot (x)) 6 x - x^{2} + \frac{d}{d x} (6 x - x^{2}) cot (x)

\frac{d}{d x} (cot (x)) = - csc^{2} (x)

\frac{d}{d x} (6 x - x^{2}) = \frac{3 - x}{6 x - x ^{2}}

= (- csc^{2} (x)) 6 x - x^{2} + \frac{3 - x}{6 x - x ^{2}} cot (x)

簡素化

(- csc^{2} (x)) 6 x - x^{2} + \frac{3 - x}{6 x - x ^{2}} cot (x) : - csc^{2} (x) 6 x - x^{2} + \frac{cot ( x ) ( 3 - x )}{6 x - x ^{2}}

= - csc^{2} (x) 6 x - x^{2} + \frac{cot ( x ) ( 3 - x )}{6 x - x ^{2}}