derivative of arctan(tanh(x))

解

\frac{d}{d x} (arctan (tanh (x)))

\frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}

解答ステップ

\frac{d}{d x} (arctan (tanh (x)))

連鎖法則を適用:

\frac{1}{( tanh ( x ) ) ^{2} + 1} \frac{d}{d x} (tanh (x))

= \frac{1}{( tanh ( x ) ) ^{2} + 1} \frac{d}{d x} (tanh (x))

\frac{d}{d x} (tanh (x)) = sech^{2} (x)

= \frac{1}{( tanh ( x ) ) ^{2} + 1} sech^{2} (x)

簡素化

\frac{1}{tanh ^{2} ( x ) + 1} sech^{2} (x) : \frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}

= \frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}