derivative of (tan^2(x)/(tan(x^2)))

解

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

2 sec^{2} (x) tan (x) cot (x^{2}) - 2 x csc^{2} (x^{2}) tan^{2} (x)

解答ステップ

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

簡素化

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

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

積の法則を適用:

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

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

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

\frac{d}{d x} (tan^{2} (x)) = 2 tan (x) sec^{2} (x)

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

= 2 tan (x) sec^{2} (x) cot (x^{2}) + (- csc^{2} (x^{2}) \cdot 2 x) tan^{2} (x)

簡素化

= 2 sec^{2} (x) tan (x) cot (x^{2}) - 2 x csc^{2} (x^{2}) tan^{2} (x)