derivative of (2x-3^{ln(x)})

解

\frac{d}{d x} ((2 x - 3)^{l n (x)})

x^{l n (2 x - 3)} (\frac{ln ( 2 x - 3 )}{x} + \frac{2 ln ( x )}{2 x - 3})

解答ステップ

\frac{d}{d x} ((2 x - 3)^{l n (x)})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(2 x - 3)^{l n (x)} = e^{l n (x) l n (2 x - 3)}

= \frac{d}{d x} (e^{l n (x) l n (2 x - 3)})

連鎖法則を適用:

e^{l n (x) l n (2 x - 3)} \frac{d}{d x} (ln (x) ln (2 x - 3))

= e^{l n (x) l n (2 x - 3)} \frac{d}{d x} (ln (x) ln (2 x - 3))

\frac{d}{d x} (ln (x) ln (2 x - 3)) = \frac{ln ( 2 x - 3 )}{x} + \frac{2 ln ( x )}{2 x - 3}

= e^{l n (x) l n (2 x - 3)} (\frac{ln ( 2 x - 3 )}{x} + \frac{2 ln ( x )}{2 x - 3})

e^{l n (x) l n (2 x - 3)} = x^{l n (2 x - 3)}

= x^{l n (2 x - 3)} (\frac{ln ( 2 x - 3 )}{x} + \frac{2 ln ( x )}{2 x - 3})