derivative y=20x^3(x^4+5)^4

解

導関数

y = 20 x^{3} (x^{4} + 5)^{4}

20 (3 x^{2} (x^{4} + 5)^{4} + 16 x^{6} (x^{4} + 5)^{3})

解答ステップ

\frac{d}{d x} (20 x^{3} (x^{4} + 5)^{4})

定数を除去:

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

= 20 \frac{d}{d x} (x^{3} (x^{4} + 5)^{4})

積の法則を適用:

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

f = x^{3}, g = (x^{4} + 5)^{4}

= 20 (\frac{d}{d x} (x^{3}) (x^{4} + 5)^{4} + \frac{d}{d x} ((x^{4} + 5)^{4}) x^{3})

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

\frac{d}{d x} ((x^{4} + 5)^{4}) = 16 x^{3} (x^{4} + 5)^{3}

= 20 (3 x^{2} (x^{4} + 5)^{4} + 16 x^{3} (x^{4} + 5)^{3} x^{3})

16 x^{3} (x^{4} + 5)^{3} x^{3} = 16 x^{6} (x^{4} + 5)^{3}

= 20 (3 x^{2} (x^{4} + 5)^{4} + 16 x^{6} (x^{4} + 5)^{3})