derivative y=(2tan^2(2x)-1)^{1/2}

解

導関数

y = (2 tan^{2} (2 x) - 1)^{\frac{1}{2}}

\frac{4 sec ^{2} ( 2 x ) tan ( 2 x )}{( 2 tan ^{2} ( 2 x ) - 1 ) ^{\frac{1}{2}}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (2 tan^{2} (2 x) - 1) = 8 sec^{2} (2 x) tan (2 x)

= \frac{1}{2 ( 2 tan ^{2} ( 2 x ) - 1 ) ^{\frac{1}{2}}} \cdot 8 sec^{2} (2 x) tan (2 x)

簡素化

\frac{1}{2 ( 2 tan ^{2} ( 2 x ) - 1 ) ^{\frac{1}{2}}} \cdot 8 sec^{2} (2 x) tan (2 x) : \frac{4 sec ^{2} ( 2 x ) tan ( 2 x )}{( 2 tan ^{2} ( 2 x ) - 1 ) ^{\frac{1}{2}}}

= \frac{4 sec ^{2} ( 2 x ) tan ( 2 x )}{( 2 tan ^{2} ( 2 x ) - 1 ) ^{\frac{1}{2}}}