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人気のある微分積分 >

derivative of cx^2e^{(-mx^2/(2kt)})

解

\frac{d}{d x} (c \cdot x^{2} \cdot e^{\frac{- m x ^{2}}{2 k t}})

解

c (- \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x ^{3}}{k t} + 2 e^{- \frac{m x ^{2}}{2 k t}} x)

解答ステップ

\frac{d}{d x} (c x^{2} e^{\frac{- m x ^{2}}{2 k t}})

簡素化

c x^{2} e^{\frac{- m x ^{2}}{2 k t}} : c e^{- \frac{m x ^{2}}{2 k t}} x^{2}

= \frac{d}{d x} (c e^{- \frac{m x ^{2}}{2 k t}} x^{2})

c, m, k, t

を定数として扱う

定数を除去:

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

= c \frac{d}{d x} (e^{- \frac{m x ^{2}}{2 k t}} x^{2})

積の法則を適用:

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

f = e^{- \frac{m x ^{2}}{2 k t}}, g = x^{2}

= c (\frac{d}{d x} (e^{- \frac{m x ^{2}}{2 k t}}) x^{2} + \frac{d}{d x} (x^{2}) e^{- \frac{m x ^{2}}{2 k t}})

\frac{d}{d x} (e^{- \frac{m x ^{2}}{2 k t}}) = - \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x}{k t}

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

= c ((- \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x}{k t}) x^{2} + 2 x e^{- \frac{m x ^{2}}{2 k t}})

簡素化

c ((- \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x}{k t}) x^{2} + 2 x e^{- \frac{m x ^{2}}{2 k t}}) : c (- \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x ^{3}}{k t} + 2 e^{- \frac{m x ^{2}}{2 k t}} x)

= c (- \frac{m e ^{- \frac{m x ^{2}}{2 k t}} x ^{3}}{k t} + 2 e^{- \frac{m x ^{2}}{2 k t}} x)

グラフ

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