derivative of (4b/a xsqrt(a^2-x^2))

解

\frac{d}{d x} (\frac{4 b}{a} x a^{2} - x^{2})

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

解答ステップ

\frac{d}{d x} (\frac{4 b}{a} x a^{2} - x^{2})

定数を除去:

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

= \frac{4 b}{a} \frac{d}{d x} (x a^{2} - x^{2})

積の法則を適用:

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

f = x, g = a^{2} - x^{2}

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

\frac{d x}{d x} = 1

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

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

簡素化

\frac{4 b}{a} (1 \cdot a^{2} - x^{2} + (- \frac{x}{a ^{2} - x ^{2}}) x) : \frac{4 b ( - 2 x ^{2} + a ^{2} )}{a a ^{2} - x ^{2}}

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