derivative f(x)=5sqrt(x)e^{x^2-3}

解

導関数

f (x) = 5 x e^{x^{2} - 3}

5 (\frac{e ^{x^{2} - 3}}{2 x} + 2 e^{x^{2} - 3} x x)

解答ステップ

\frac{d}{d x} (5 x e^{x^{2} - 3})

定数を除去:

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

= 5 \frac{d}{d x} (x e^{x^{2} - 3})

積の法則を適用:

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

f = x, g = e^{x^{2} - 3}

= 5 (\frac{d}{d x} (x) e^{x^{2} - 3} + \frac{d}{d x} (e^{x^{2} - 3}) x)

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

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

= 5 (\frac{1}{2 x} e^{x^{2} - 3} + e^{x^{2} - 3} \cdot 2 x x)

\frac{1}{2 x} e^{x^{2} - 3} = \frac{e ^{x^{2} - 3}}{2 x}

= 5 (\frac{e ^{x^{2} - 3}}{2 x} + 2 e^{x^{2} - 3} x x)