derivative g(x)= 1/(4x^2+2x)

解

導関数

g (x) = \frac{1}{4 x ^{2} + 2 x}

- \frac{4 x + 1}{2 x ^{2} ( 2 x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1}{4 x ^{2} + 2 x})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((4 x^{2} + 2 x)^{- 1})

連鎖法則を適用:

- \frac{1}{( 4 x ^{2} + 2 x ) ^{2}} \frac{d}{d x} (4 x^{2} + 2 x)

= - \frac{1}{( 4 x ^{2} + 2 x ) ^{2}} \frac{d}{d x} (4 x^{2} + 2 x)

\frac{d}{d x} (4 x^{2} + 2 x) = 8 x + 2

= - \frac{1}{( 4 x ^{2} + 2 x ) ^{2}} (8 x + 2)

簡素化

- \frac{1}{( 4 x ^{2} + 2 x ) ^{2}} (8 x + 2) : - \frac{4 x + 1}{2 x ^{2} ( 2 x + 1 ) ^{2}}

= - \frac{4 x + 1}{2 x ^{2} ( 2 x + 1 ) ^{2}}