derivative of 2sqrt(x)tanh(sqrt(x))

解

\frac{d}{d x} (2 x tanh (x))

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

解答ステップ

\frac{d}{d x} (2 x tanh (x))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

f = x, g = tanh (x)

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

\frac{d}{d x} (x) = \frac{1}{2 x}

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

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

簡素化

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

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