derivative of 1+(5e^{-2x^2}/(2-|x|))

解

\frac{d}{d x} (1 + \frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣})

\frac{5 ( e ^{- 2 x^{2}} x - 4 e ^{- 2 x^{2}} x ∣ x ∣ ( - ∣ x ∣ + 2 ) )}{∣ x ∣ ( - ∣ x ∣ + 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (1 + \frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣})

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (1) + \frac{d}{d x} (\frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣})

\frac{d}{d x} (1) = 0

\frac{d}{d x} (\frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣}) = \frac{5 ( - 4 e ^{- 2 x^{2}} x ∣ x ∣ ( 2 - ∣ x ∣ ) + e ^{- 2 x^{2}} x )}{∣ x ∣ ( 2 - ∣ x ∣ ) ^{2}}

= 0 + \frac{5 ( - 4 e ^{- 2 x^{2}} x ∣ x ∣ ( 2 - ∣ x ∣ ) + e ^{- 2 x^{2}} x )}{∣ x ∣ ( 2 - ∣ x ∣ ) ^{2}}

簡素化

= \frac{5 ( e ^{- 2 x^{2}} x - 4 e ^{- 2 x^{2}} x ∣ x ∣ ( - ∣ x ∣ + 2 ) )}{∣ x ∣ ( - ∣ x ∣ + 2 ) ^{2}}