derivative of (1-x/2 ^{e^{x^2}})

解

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

(2 e^{x^{2}} x ln (1 - \frac{x}{2}) - \frac{e ^{x^{2}}}{2 - x}) (1 - \frac{x}{2})^{e^{x^{2}}}

解答ステップ

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

指数の規則を適用する:

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

(1 - \frac{x}{2})^{e^{x^{2}}} = e^{e^{x^{2}} l n (1 - \frac{x}{2})}

= \frac{d}{d x} (e^{e^{x^{2}} l n (1 - \frac{x}{2})})

連鎖法則を適用:

e^{e^{x^{2}} l n (1 - \frac{x}{2})} \frac{d}{d x} (e^{x^{2}} ln (1 - \frac{x}{2}))

= e^{e^{x^{2}} l n (1 - \frac{x}{2})} \frac{d}{d x} (e^{x^{2}} ln (1 - \frac{x}{2}))

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

= e^{e^{x^{2}} l n (1 - \frac{x}{2})} (2 e^{x^{2}} x ln (1 - \frac{x}{2}) - \frac{e ^{x^{2}}}{2 - x})

e^{e^{x^{2}} l n (1 - \frac{x}{2})} = (1 - \frac{x}{2})^{e^{x^{2}}}

= (2 e^{x^{2}} x ln (1 - \frac{x}{2}) - \frac{e ^{x^{2}}}{2 - x}) (1 - \frac{x}{2})^{e^{x^{2}}}