derivative of sec^2(x/a+csc^2(x/a))

解

\frac{d}{d x} (sec^{2} (\frac{x}{a}) + csc^{2} (\frac{x}{a}))

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

解答ステップ

\frac{d}{d x} (sec^{2} (\frac{x}{a}) + csc^{2} (\frac{x}{a}))

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (sec^{2} (\frac{x}{a})) + \frac{d}{d x} (csc^{2} (\frac{x}{a}))

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

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

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

規則を適用

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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