derivative of 1/2 x^2sqrt(36-x^2)

解

\frac{d}{d x} (\frac{1}{2} x^{2} 36 - x^{2})

\frac{72 x - 3 x ^{3}}{2 36 - x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1}{2} x^{2} 36 - x^{2})

定数を除去:

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

= \frac{1}{2} \frac{d}{d x} (x^{2} 36 - x^{2})

積の法則を適用:

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

f = x^{2}, g = 36 - x^{2}

= \frac{1}{2} (\frac{d}{d x} (x^{2}) 36 - x^{2} + \frac{d}{d x} (36 - x^{2}) x^{2})

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

\frac{d}{d x} (36 - x^{2}) = - \frac{x}{36 - x ^{2}}

= \frac{1}{2} (2 x 36 - x^{2} + (- \frac{x}{36 - x ^{2}}) x^{2})

簡素化

\frac{1}{2} (2 x 36 - x^{2} + (- \frac{x}{36 - x ^{2}}) x^{2}) : \frac{72 x - 3 x ^{3}}{2 36 - x ^{2}}

= \frac{72 x - 3 x ^{3}}{2 36 - x ^{2}}