derivative of sqrt({g)(x}*(ax^2+x))

解

\frac{d}{d x} (g (x) \cdot (a x^{2} + x))

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

解答ステップ

\frac{d}{d x} (g (x) (a x^{2} + x))

積の法則を適用:

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

f = g (x), g = a x^{2} + x

= \frac{d}{d x} (g (x)) (a x^{2} + x) + \frac{d}{d x} (a x^{2} + x) g (x)

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

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

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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