derivative of 5(x^2+7^4(2x))

解

\frac{d}{d x} (5 (x^{2} + 7)^{4} (2 x))

10 ((x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3})

解答ステップ

\frac{d}{d x} (5 (x^{2} + 7)^{4} (2 x))

簡素化

5 (x^{2} + 7)^{4} (2 x) : 10 x (x^{2} + 7)^{4}

= \frac{d}{d x} (10 x (x^{2} + 7)^{4})

定数を除去:

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

= 10 \frac{d}{d x} (x (x^{2} + 7)^{4})

積の法則を適用:

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

f = x, g = (x^{2} + 7)^{4}

= 10 (\frac{d x}{d x} (x^{2} + 7)^{4} + \frac{d}{d x} ((x^{2} + 7)^{4}) x)

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

\frac{d}{d x} ((x^{2} + 7)^{4}) = 8 x (x^{2} + 7)^{3}

= 10 (1 \cdot (x^{2} + 7)^{4} + 8 x (x^{2} + 7)^{3} x)

簡素化

10 (1 \cdot (x^{2} + 7)^{4} + 8 x (x^{2} + 7)^{3} x) : 10 ((x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3})

= 10 ((x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3})