derivative of 3x^2sqrt(25-x^2)

解

\frac{d}{d x} (3 x^{2} 25 - x^{2})

\frac{3 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}

解答ステップ

\frac{d}{d x} (3 x^{2} 25 - x^{2})

定数を除去:

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

= 3 \frac{d}{d x} (x^{2} 25 - x^{2})

積の法則を適用:

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

f = x^{2}, g = 25 - x^{2}

= 3 (\frac{d}{d x} (x^{2}) 25 - x^{2} + \frac{d}{d x} (25 - x^{2}) x^{2})

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

\frac{d}{d x} (25 - x^{2}) = - \frac{x}{25 - x ^{2}}

= 3 (2 x 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) x^{2})

簡素化

3 (2 x 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) x^{2}) : \frac{3 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}

= \frac{3 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}