derivative of 4x^{0.75}*(240-2x^{0.5})

解

\frac{d}{d x} (4 x^{0.75} \cdot (240 - 2 x)^{0.5})

\frac{4 ( 180 - 2.5 x )}{x ^{0.25} ( - 2 x + 240 ) ^{\frac{1}{2}}}

解答ステップ

\frac{d}{d x} (4 x^{0.75} (240 - 2 x)^{0.5})

定数を除去:

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

= 4 \frac{d}{d x} (x^{0.75} (240 - 2 x)^{0.5})

積の法則を適用:

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

f = x^{0.75}, g = (240 - 2 x)^{0.5}

= 4 (\frac{d}{d x} (x^{0.75}) (240 - 2 x)^{0.5} + \frac{d}{d x} ((240 - 2 x)^{0.5}) x^{0.75})

\frac{d}{d x} (x^{0.75}) = \frac{0.75}{x ^{0.25}}

\frac{d}{d x} ((240 - 2 x)^{0.5}) = - \frac{1}{( 240 - 2 x ) ^{\frac{1}{2}}}

= 4 (\frac{0.75}{x ^{0.25}} (240 - 2 x)^{0.5} + (- \frac{1}{( 240 - 2 x ) ^{\frac{1}{2}}}) x^{0.75})

簡素化

4 (\frac{0.75}{x ^{0.25}} (240 - 2 x)^{\frac{1}{2}} + (- \frac{1}{( 240 - 2 x ) ^{\frac{1}{2}}}) x^{0.75}) : \frac{4 ( 180 - 2.5 x )}{x ^{0.25} ( - 2 x + 240 ) ^{\frac{1}{2}}}

= \frac{4 ( 180 - 2.5 x )}{x ^{0.25} ( - 2 x + 240 ) ^{\frac{1}{2}}}