derivative of sqrt(5)x^5{f}(x)

解

\frac{d}{d x} (5 x^{5} f (x))

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

解答ステップ

\frac{d}{d x} (5 x^{5} f (x))

定数を除去:

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

= 5 \frac{d}{d x} (x^{5} f (x))

積の法則を適用:

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

f = x^{5}, g = f (x)

= 5 (\frac{d}{d x} (x^{5}) f (x) + \frac{d}{d x} (f (x)) x^{5})

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

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